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INFERENTIAL MODEL PREDICTIVE CONTROL OF

POLY(ETHYLENE TEREPHTHALATE) DEGRADATION DURING EXTRUSION

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

MURAT OLUŞ ÖZBEK

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

CHEMICAL ENGINEERING

AUGUST 2006

2

Approval of the Graduate School of Natural and Applied Sciences.

___________________________

Prof Dr. Canan Özgen

Director

I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of

Science.

___________________________

Prof. Dr. Nurcan Baç

Head of Department

This is to certify that we have read this thesis and that in our opinion it is fully adequate, in

scope and quality, as a thesis for the degree of Master of Science.

__________________________ __________________________

Assoc. Prof. Dr. Göknur Bayram Prof. Dr. Canan Özgen

Co-supervisor Supervisor

Examining Committee Members

Prof. Dr. Güngör Gündüz

(METU, CHE)________________________

Prof. Dr. Canan Özgen

(METU, CHE)________________________

Doç Dr. Göknur Bayram

(METU, CHE)________________________

Doç. Dr. Naime Aslı Sezgi

(METU, CHE)________________________

Prof. Dr. Kemal Leblebicioğlu

(METU, EE)________________________

iii

PLAGIARISM

I hereby declare that all information in this document has been obtained and presented in

accordance with academic rules and ethical conduct. I also declare that, as required by

these rules and conduct, I have fully cited and referenced all material and results that are

not original to this work.

Name, Last name:

Signature :

iv

ABSTRACT

INFERENTIAL MODEL PREDICTIVE CONTROL OF POLY(ETHYLENE TEREPHTHALATE) DEGRADATION

DURING EXTRUSION

Özbek, Murat Oluş

M. S., Department of Chemical Engineering

Supervisor: Prof. Dr. Canan Özgen

Co-Supervisor: Assoc. Prof. Dr. Göknur Bayram

August 2006, 91 pages

Poly(ethylene terephthalate), PET, which is commonly used as a packaging material, is not

degradable in nature. As an issue of sustainable development it must be recycled and

converted into other products. During this process, extrusion is an important unit operation.

In extrusion process, if the operating conditions are not controlled, PET can go under

degradation, which results in the loss of some mechanical properties.

In order to overcome the degradation of recycled PET (RPET), this study aims the control of

the extrusion process. Dynamic models of the system for control purposes are obtained by

experimental studies. In the experimental studies, screw speed, feed rate and barrel

temperatures are taken as process variables in the ranges of 50 – 500 rpm, 3.85 – 8.16

g/min and 270 – 310 oC respectively. Singular value decomposition (SVD) technique is used

for the best pairing between the manipulated – controlled variables, where screw speed is

taken as the manipulated variable and molecular weight of the product is taken as the

controlled variable. PID and model predictive controller (MPC) are designed utilizing the

dynamic models in the feedback inferential control algorithm. In the simulation studies, the

performance of the designed inferential control system, where molecular weight (Mv) of the

product is estimated from the measured intrinsic viscosity ([η]) of the product, is

investigated.

v

The controller utilizing PID and MPC control algorithms are found to be robust and

satisfactory in tracking the given set points and eliminating the effects of the disturbances.

Keywords: extrusion, modeling, MPC, inferential control, PET recycling.

vi

ÖZ

EKSTRÜZYON İŞLEMİNDE

POLY(ETİLEN TEREFTALAT) BOZUNMASININ

ALGISAL MODEL ÖNGÖRÜMLÜ DENETLEÇ İLE DENETLENMESİ

Özbek, Murat Oluş

Yüksek Lisans, Kimya Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. Canan Özgen

Ortak Tez Yöneticisi: Doç. Dr. Göknur Bayram

Ağustos 2006, 91 sayfa

Genellikle, paketleme malzemesi olarak kullanılan polietilen tereftalat’ın (PET) doğada bozunma özelliği

yoktur. Bu nedenle, sürdürülebilir gelişme sürecinde PET’in geri kazanılması ve yararlı ürünlere

dönüştürülmesi gerekir. Bu işleme süreçlerinde ekstrüzyon yöntemi önemli bir temel işlemdir. Ancak,

ekstrüzyon sürecinde koşulların iyi ayarlanamadığı durumlarda, PET, bozunmaya (degradation)

uğrayabilir ve bunun sonucunda mekanik bazı özelliklerini kaybedebilir.

Bu çalışmada, PET’in bozunmasının önlenebilmesi için ekstrüzyon sürecinin denetimi amaçlanmıştır.

Denetimde kullanılmak üzere, sistemin dinamik modelleri deneysel çalışmalar yardımıyla elde edilmiştir.

Deneysel çalışmalarda süreç değişkenleri olarak 270 - 310 oC aralığında kovan sıcaklığı (T), 50 – 500

dev/dak aralığında vida hızı (SS), 3.85 - 8.16 g/dak aralığında besleme oranı (FR) seçilmiştir. En iyi

ayarlanan-denetlenen değişken eşleştirmesi için Tekil Değer Ayrıştırma (SVD) tekniği kullanılmış ve

ayarlanan değişken olarak vida hızı (SS), denetlenen değişken olarak ürünün molekül ağırlığı (Mv)

seçilmiştir. Elde edilen dinamik modellerle geleneksel (PID) ve model öngörümlü denetleçler (MÖD)

tasarlanmış ve geri beslemeli, algısal denetim algoritmasında kullanılmıştır. Benzetim çalışmalarında,

ürünün molekül ağırlığının (Mv) ölçülen içsel viskozite ([η]) değerlerinden tahmin edildiği algısal

denetim yapısının performansı incelenmiştir.

PID ve MÖD denetim algoritmaları kullanan denetleçlerin hem gürbüz hem de ayar noktası izleme ve

bozan etkenin etkisini uzaklaştırma konularında başarılı oldukları irdelenmiştir.

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Anahtar Sözcükler: ekstrüzyon, modelleme, model öngörümlü denetim, algısal denetim, geri

dönüşümlü PET.

viii

Dedicated to all my beloved ones…

ix

ACKNOWLEDGMENTS

First of all, I wish to express my sincere gratitude to my supervisor Prof. Dr. Canan Özgen

and co-supervisor Assoc. Prof. Göknur Bayram for their valuable suggestions, guidance and

understanding throughout my graduate study.

I would like to thank to all of my colleagues; especially to Salih Obut, Almıla Bahar, İsmail

Doğan, Hatice Ceylan, Sertan Yeşil, Güralp Özkoç, Işıl Işık, Özcan Köysüren and Mert Kılınç

for their help and support during my graduate study.

I would like to thank to all my co-workers in Middle East Technical University Graduate

School of Natural and Applied Sciences for their support and understanding.

Last but not least, I wish to express my deepest thanks to my family for their great support,

encouragement, and patience throughout my all education.

x

TABLE OF CONTENTS

PLAGIARISM ................................................................................................................... iii

ABSTRACT......................................................................................................................iv

ÖZ .................................................................................................................................vi

ACKNOWLEDGMENTS......................................................................................................ix

TABLE OF CONTENTS...................................................................................................... x

LIST OF TABLES............................................................................................................. xii

LIST OF FIGURES.......................................................................................................... xiv

NOMENCLATURE.......................................................................................................... xvii

CHAPTER

1. INTRODUCTION.......................................................................................................... 1

2. LITERATURE SURVEY .................................................................................................. 3

2.1 Extrusion, Extruder Modeling and Extruder Control................................................... 3

2.2 PET Recycling and PET Degradation ........................................................................ 8

2.3 Model Predictive Control (MPC) ..............................................................................10

2.4 Inferential Control .................................................................................................12

2.5 Singular Value Decomposition (SVD) ......................................................................13

3. BACKGROUND INFORMATION ON PET AND CONTROL TECHNIQUES ............................14

3.1 Poly(ethylene terephthalate) (PET).........................................................................14

3.1.1 Recycling........................................................................................................15

3.1.2 Degradation....................................................................................................16

3.2 Control Techniques ...............................................................................................18

3.2.1 MPC ...............................................................................................................18

3.2.1.1 MPC Algorithm..........................................................................................18

3.2.1.2 Constrained MPC ......................................................................................20

3.2.3 Inferential Control ..............................................................................................21

3.3 Singular Value Decomposition (SVD) ......................................................................22

4. EXPERIMENTAL STUDIES............................................................................................24

4.1 Properties of RPET and Trifluoroacetic Acid (TFA) ...................................................24

4.2 Experimental Setup ...............................................................................................25

4.3 Experimental Procedure.........................................................................................27

xi

4.3.1 Preliminary Experiments ..................................................................................27

4.3.2 Steady State Experiments................................................................................28

4.3.3 Dynamic Experiments......................................................................................29

5. RESULTS AND DISCUSSIONS......................................................................................31

5.1 Preliminary Experimental Results............................................................................31

5.2 Steady State Experimental Results .........................................................................33

5.3 Dynamic Experimental Results ...............................................................................36

5.4 Modeling Studies...................................................................................................39

5.4.1 Modeling Technique 1: Single Transfer Functions..............................................41

5.4.2 Modeling Technique 2: Dual Transfer Functions ................................................44

5.4.3 Modeling Technique 3: Discrete Convolution Models..........................................46

5.5 Control Studies .....................................................................................................49

5.5.1 Singular Value Decomposition..........................................................................49

5.5.2 Design of Control Scheme ...............................................................................51

5.5.3 PID Control.....................................................................................................52

5.5.4 Model Predictive Control (MPC) ........................................................................54

5.5.4.1 Process Models for MPC ............................................................................54

5.5.4.2 MPC Tuning..............................................................................................55

5.5.5 Robustness Analysis ........................................................................................63

6. CONCLUSIONS...........................................................................................................72

REFERENCES .................................................................................................................73

APPENDICES

A. MOLECULAR WEIGHT DETERMINATION......................................................................77

B. CALIBRATION FOR RESIDENCE TIME AND FEED RATE.................................................79

C. EXPERIMENTAL DATA ................................................................................................82

D. PID CONTROL LOOP AND PROGRAM CODES ...............................................................84

E. SAMPLE CALCULATIONS.............................................................................................88

xii

LIST OF TABLES

TABLES

Table 3.1: Minimum requirements for RPET flakes to be reprocessed [Ajawa and Pavel,

2005]. ....................................................................................................................16

Table 4.1: Properties of RPET resin (AdvanSA).................................................................25

Table 4.2: Step changes given to the process variables. ...................................................29

Table 5.1: Measured intrinsic viscosity and molecular weight of unprocessed RPET. ...........32

Table 5.2: Results of consistency check ...........................................................................36

Table 5.3: Single transfer function models and their normalized IAE scores. ......................42

Table 5.4: Dual transfer function models and their normalized IAE scores. ........................44

Table 5.5: PID settings utilizing modified Z-N method (Some Overshoot) [Seborg et al.,

1989], and normalized IAE scores of set point tracking (Figure 5.19)..........................52

Table 5.6: Fine tuned PID settings for PID+ and PID-........................................................53

Table 5.7: The magnitudes and the time of changes of the given set point changes...........54

Table 5.8: MPC tuning parameters. .................................................................................56

Table 5.9: Normalized IAE scores of controllers throughout set point tracking....................57

Table 5.10: Normalized IAE scores and settling times of controllers in disturbance rejection

performances. .........................................................................................................60

Table 5.11: Normalized IAE scores of controllers throughout set point tracking (10%

increase in model gains). .........................................................................................65


decrease in model gains). ........................................................................................65


performances (10% increase in model gains). ...........................................................70


performances (10% decrease in models gains). ........................................................71

Table B.1: Average residence times for studied screw speeds. ..........................................79

Table B.2: Flow rate calibration data for studied feed rate settings....................................80

Table C.1: Molecular weight (g/gmol), Mv, change for different SS and FR values at 270 oc.

..............................................................................................................................82


..............................................................................................................................82

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..............................................................................................................................82

Table C.4: Mv (g/gmol) change with SS, FR and T (dynamic runs).....................................83

Table E.1: Experimental data (an example)......................................................................88

Table E.2: Reduced viscosity for different concentrations..................................................89

Table E.3: Ultimate gains and ultimate periods of positive and negative PID-model pairs....90

Table E.4: Original and modified Ziegler-Nichols settings for PID controllers [Seborg et al.,

1989]. ....................................................................................................................91

xiv

LIST OF FIGURES

FIGURES

Figure 3.1: PET synthesis reactions: (a) trans-esterification reaction and (b) condensation

reaction [Ajawa and Pavel, 2005]. ............................................................................15

Figure 3.2: Thermal degradation of PET [Karayannidis et al., 2000]. .................................17

Figure 3.3: Acid alcohol condensation of PET [Karayannidis and Psalida, 2000]..................17

Figure 3.4: Open loop step response of a linear plant [Seborg et al., 1989]. ......................18

Figure 3.5: Block diagram for inferential control loop........................................................21

Figure 4.1: Molecular formula of trifluoroacetic acid (TFA). ...............................................24

Figure 4.2: Extruder used for experiments. ......................................................................26

Figure 4.3: Schematic drawing for the experimental setup. ...............................................26

Figure 4.4: Control panel of the extruder (a: die temperature and pressure, b: melt

temperatures, c: barrel temperatures, d: screw speed and torque, e: main feed rate). 27

Figure 5.1: The effect of temperature on molecular weight of RPET. .................................32

Figure 5.2: Effect of SS and FR on Mv at T = 270 oC.........................................................34



Figure 5.5: System responses to (a) positive (+25 rpm) and (b) negative (-50 rpm) steps on

screw speed............................................................................................................37

Figure 5.6: System responses to (a) positive (+1.04 g/min) (+25) and (b) negative (-1.42

g/min) (-25) steps on feed rate. ...............................................................................38

Figure 5.7: System responses to (a) positive (+20 oC) and (b) negative (-20 oC) steps on

temperature. ...........................................................................................................38

Figure 5.8: Model representation of the system................................................................40

Figure 5.9: Plant and model responses to a (a) positive and (b) negative unit step changes

in SS (single transfer function case)..........................................................................41


in FR (single transfer function case)..........................................................................43


in T (single transfer function case). ..........................................................................43

xv


in SS (dual transfer functions case). .........................................................................45


in FR (dual transfer functions case). .........................................................................45


in T (dual transfer functions case). ...........................................................................46


in SS (discrete convolution models case)...................................................................47


in FR (discrete convolution models case)...................................................................48


in T (discrete convolution models case). ...................................................................48

Figure 5.18: Designed control system..............................................................................52

Figure 5.19: Set point tracking of PID controllers of (a) positive and (b) negative models,

tuned using modified Z-N settings (Some Overshoot) [Seborg et al., 1989].................53

Figure 5.20: Responses of ARX models to (a) positive and (b) negative unit step changes in

SS. .........................................................................................................................55

Figure 5.21: Set point tracking responses of PID and MPC. ...............................................56

Figure 5.22: Disturbance rejection responses of PID and MPC (+25 or +1.04 g/gmol step

disturbance on FR). .................................................................................................57

Figure 5.23: Disturbance rejection responses of PID and MPC (-25 or -1.42 g/gmol step


Figure 5.24: Disturbance rejection responses of PID and MPC (+10 oC step disturbance on

T)...........................................................................................................................59

Figure 5.25: Disturbance rejection responses of PID and MPC (-10 oC step disturbance on T).

..............................................................................................................................59

Figure 5.26: Disturbance rejection responses of PID and MPC (+12.5 or +0.57 g/gmol step


Figure 5.27: Disturbance rejection responses of PID and MPC (-12.5 or -0.71 g/gmol step


Figure 5.28: Disturbance rejection responses of PID and MPC (+20 oC step disturbance on

T)...........................................................................................................................62


..............................................................................................................................62

Figure 5.30: Set point tracking responses of PID and MPC (10% increase in model gains). .64

Figure 5.31: Set point tracking responses of PID and MPC (10% decrease in model gains).64

xvi

Figure 5.32: Disturbance rejection responses of PID and MPC (+1.04 g/gmol step

disturbance on FR) (10% increase in model gains). ...................................................66

Figure 5.33: Disturbance rejection responses of PID and MPC (-1.42 g/gmol step disturbance

on FR) (10% increase in model gains). .....................................................................66


disturbance on FR) (10% decrease in model gains). ..................................................67

Figure 5.35: Disturbance rejection responses of PID and MPC (-1.42 g/gmol step disturbance

on FR) (10% decrease in model gains). ....................................................................67

Figure 5.36: Disturbance rejection responses of PID and MPC (+10 oC step disturbance on T)

(10% increase in model gains). ................................................................................68

Figure 5.37: Disturbance rejection responses of PID and MPC (-10 oC step disturbance on T)

(10% increase in model gains). ................................................................................69


(10% decrease in model gains). ...............................................................................69


(10% decrease in model gains). ...............................................................................70

Figure A.1: Ostwald and Ubbelohde viscometers. .............................................................78

Figure B.1: Avarage residence times measured at different screw speeds. .........................80

Figure B.2: Feed rate settings and corresponding flow rates (g/min). ................................81

Figure D.1: Feedback control loop for PID controllers designed in SIMULINK. ....................84

Figure E.1: Reduced viscosity change with concentration..................................................90

xvii

NOMENCLATURE

a Step response coefficient

A Coefficient matrix

E Error

E Error vector

FR Feed Rate (g/gmol)

G Steady-state gain matrix

h Impulse response coefficient

K Gain matrix

K Gain value or Mark-Hauwing constant

L Control horizon

m Manipulated variable

M Input vector

Mv Viscosity average molecular weight

Mw Weight average molecular weight

PP Polypropylene

r Set point

R Prediction horizon

SS Screw Speed (rpm)

T Temperature (oC)

U Left singular matrix in SVD

V Right singular matrix in SVD

y Output

Y Output vector

Subscripts:

n Sample number (in means of ∆t)

Superscripts:

c Corrected prediction

d Desired value

p Predicted value

N Normalized value

xviii

Greek Letters:

α Filter value or Mark-Hauwing constant

λ Weighting matrix diagonal element

τ Process transfer function time constant

∆ Increment

ω Weighting matrices

[ ]η Intrinsic viscosity

redη Reduced viscosity

relη Relative viscosity

Σ Sum or Singular value matrix in SVD

Abbreviations:

ANN Artificial Neural Network

ARTD Average Residence Time Distribution

DMC Dynamic Matrix Control

ILR In-line reometer

IMC Internal Model Control

IV Intrinsic Viscosity

LDPE Low Density Polyethylene

MAC Model Algorithmic Control

MFI Melt Flow Index

MI Melt Index

MIMO Multiple Input Multiple Output

MPC Model Predictive Control

MPHC Model Predictive Heuristic Control

MVC Minimum Variance Conroller

PET Poly(ethylene terephthalate)

PVC Polyvinyl chloride

RPET Recycled poly(ethylene terephthalate)

RTD Residence Time Distribution

SISO Single Input Single Output

SSE Single Screw Extruder

STR Self Tuning Regulator

SVD Singular Value Decomposition

TFA Trifluoroacetic acid

TSE Twin Screw Extruder

1

CHAPTER I

INTRODUCTION

PET, which is a thermoplastic resin of polyester family, has become one of the major

packaging materials due to its good barrier and mechanical properties. Due to its non-

degradable nature, PET must be recycled for sustainable development and must be used to

obtain other byproducts. Today, PET is one of the most recycled materials world wide.

The major drawback in PET recycling is the loss of the molecular weight, a phenomenon

known as degradation. Heat, mechanical effects, contaminants and water moisture are the

important factors of degradation. Degradation has an adverse effect on the mechanical

properties of the product. Therefore, degradation amount should be reduced to preserve the

mechanical properties. Molecular weight can be used to express the degradation amount.

Extruders are commonly used machinery in plastics processing and recycling industries.

Previous studies [Incarnato et al., 2000; Spinace et al., 2000; Assadi et al., 2004; Ajawa and

Pawel, 2004] showed that the degradation of RPET is caused mainly by processes taking

place in the extruder. Furthermore, the quality of an extruded product is directly related to

its rheological properties such as melt viscosity of the material in the extruder. Thus,

extrusion process should be controlled in order to reduce the degradation amount.

Previous studies [Parnaby et al., 1975; Smith et al., 1978; Hassan and Parnaby, 1981, Costin

et al., 1982; Yang and Lee, 1986; Tanttu et al., 1989; Pabendiskas et al., 1989; Nied et al.,

2000; Xiao et al., 2001; Previdi et al., 2006] on the control of an extruder or extrusion

process mainly focused on regulating the process parameters like screw speed, barrel

temperatures or barrel/die temperature or pressure on defined preset values. However, in

such a regulation, expert knowledge on the operation and on the relations between the

process conditions and product properties is required. In order to eliminate the requirement

of such an expert knowledge, by using the secondary measurements, an inferential control

2

scheme can be designed for the control of desired product property whose online

measurement is not possible.

Product property is important for the downstream operations of the extruder and can be

kept constant at a desired point by designing a proper control mechanism for the extruder.

Among the different types of controllers, model predictive controller (MPC) has proven itself

worthy and being used in the industry for the last two decades. Although different

algorithms exist for MPC, the name basically refers to a family of controllers in which the

future behavior of the plant is predicted using a dynamic model of the plant and necessary

control actions are calculated in an optimal manner.

In this study, the control of product quality in terms of molecular weight of extruded RPET is

aimed. The system parameters affecting the molecular weight (Mv) are screw speed (SS),

feed rate (FR) and temperature (T). Thus the inputs are considered to be SS, FR and T, and

the output as Mv. The die temperature, die pressure and torque which are additional

parameters are not taken as variables in the scope of this study. Nominal operating point in

term of SS, FR and T, suitable for fiber production is determined by using the steady state

experimental data. Dynamic experiments are conducted to obtain the relations between the

inputs (SS, FR and T) and the output (molecular weight of the extruded product) of the

extruder. These data are used to obtain the dynamic models of the process. An inferential

control scheme is designed to control the molecular weight of the product for different

disturbances by simulation studies by using dynamic models. In the designed control

scheme, performances of PID and MPC controllers are studied.

The outline of this work is as follows. A literature survey on previous studied on related

subjects is given in Chapter 2. Background information about PET and control techniques are

given in Chapter 3. Experimental studies including the set-up and procedures are

summarized in Chapter 4. Chapter 5 includes the results of all the experimental, modeling

and control studies with discussions. Conclusions are given in the last chapter, Chapter 6.

3

CHAPTER II

LITERATURE SURVEY

PET is one of the commonly recycled industrial plastics. However, in order to overcome the

phenomenon of loosing material property during recycling, one option is to control the

process to obtain a product with desired property. Control of product property in an

extrusion process is a challenging one as the aimed product property cannot be measured

online in most of the cases. This problem can be overcome by building an inferential control

scheme by measuring the viscosity of the product on-line and predicting the molecular

weight from this measurement. Furthermore, using a model based controller seems

promising to yield more successful results. A literature survey about these subjects is

presented in this chapter.

2.1 Extrusion, Extruder Modeling and Extruder Control

Parnaby et al. (1975) studied the automatic control of an extruder. Their work mainly

outlines the basics of the development of a feed-forward adaptive predictive control

strategy. Also system identification and modeling were done such as stochastic identification

techniques, step and impulse models. The basic structure for the extruder control scheme

and the interaction between the extruder and the die variables were illustrated. In their

work screw speed was accepted as the manipulated variable and the die pressure, as a

measure of degree of mixing, which indirectly gives the product quality, as the controlled

variable. The melt temperature was also monitored to understand the dynamics better. It is

also stated that by on-line updating the model, small changes in model can be compensated.

Smith et al. (1978) explained the operating characteristics of twins screw extruders (TSE).

The study focused on the interrelationships between the design parameters (such as screw

design, die geometry, feed zone geometry), material properties, and operation variables

4

(such as screw speed, barrel and die temperatures). The effects of these variables and

parameters on the product and product quality during operation are also reviewed. In the

work, first, the effects of the parameters were explained mathematically under certain

assumptions. Finally, these parameter based mathematical models were illustrated for

control purposes. Neither experimental nor simulation studies were carried out. Only the

responses to changes in screw speed and feed rate were illustrated using the obtained

models.

Hassan and Parnaby (1981) experimentally constructed a cascade (hierarchical) control loop

on a laboratory scale extruder and studied the feed-back and feed-forward control

strategies. The constructed system measured barrel and die wall temperatures and

pressures, screw speed and the ‘restrictor valve’ position on the die. Measuring these, the

control system sent set points to screw speed, die restrictor valve, barrel and die wall

temperatures. To achieve this goal, the controller used the constructed model of the

extruder and die behavior, and an optimization function to calculate the control action. Also,

in the work two aims of the extruder control are specified as set point tracking (steady-state

control) and disturbance rejection (dynamic control).

Costin et al. (1982) published a review on the present literature about the dynamic modeling

and control of plasticizing extruder. They developed the subject into three headings, which

were extruder disturbance studies, classical control studies and stochastic control studies. It

was pointed that the previous works were mainly focused on long-term disturbances related

with the melt temperature, melt pressure or extrudate thickness.

Costin et al. (1982) studied the effects of the screw speed on the die pressure and

temperature. System dynamics were modeled as first order with dead time and time series

models. Die pressure was controlled by manipulating screw speed. The disturbance was

introduced to the system as difference in feed material composition. Digital PI, self tuning

regulator (STR) and minimum variance controllers (MVC) were studied and compared. The

results of the study showed that instead of eliminating the disturbances, the STR tuned itself

to eliminate the flight noise, which is caused due to the rotation of the screw.

Yang and Lee (1986) proposed several feedback and feed-forward control methods to

control long term and short term disturbances and evaluated these methods using various

load changes. The aim was to control the extrudate thickness by manipulating the take-up

speed against the screw speed load. A first order model was developed and PI controller

5

with Smith predictor and digital noise filters were tested for load and set-point changes. The

experimental results showed that the control structure was good at set-point changes but its

performance was low at load changes due to the inaccuracy of the model.

Tanttu et al. (1989) planted STR on extruder in simulation level in order to control the start-

up period of the extruder by regulating the barrel temperatures. For mathematical

description of the system a distributed parameter model was used. For parameter estimation

different algorithms were tested. The simulation results showed that STR can be used with a

proper parameter estimation algorithm for the start-up and normal operation.

Pabendinskas et al. (1989) studied the degradation of polypropylene (PP) in a reactive

extrusion process to produce PP with a specified molecular weight (Mw). The measured and

manipulated variables were die pressure drop and initiator concentration, respectively. The

initiator (peroxide) concentration was manipulated via syringe pump. Online measured

viscosity via die pressure drop, and melt flow index (MFI), measured via online rheometer,

were used to determine the amount of degradation. Step tests were performed to find the

relation between the die pressure drop and degradation, and this relation was modeled with

a first order with dead time model. The implemented controller scheme was consisted of a

gain scheduling controller with Smith dead time compensator and a PI controller. The PI

controller was used in a cascade manner to control the syringe pump. The results of the

study indicated that the desired Mw could be achieved by the control of the pressure drop.

Boadhead et al. (1996) developed an in-line rheometer (ILR), which had a ‘partial Couette’

geometry, to reduce the measurement delay, and described its use. The reactive extrusion

process was modeled as a first order with dead time process. PI and minimum variance

controllers were tested. The results showed that there was a considerable dead time in spite

of the ILR advantages. It is also offered that the use of adaptive techniques for such a

system could improve the controller performance, because the tested controllers’

performances were not good due to the non-linearity of the process.

De Ruyck (1997) developed a residence time distribution (RTD) model for a twin screw food

extruder. The model was constructed as series of CSTRs with recycling flows and different

volumes. Effects of the variables (screw profile, screw speed, water supply, feed supply and

die diameter) on RTD were observed experimentally and compared with the model results

and seen to be in agreement. By using the model the effect of different screw designs were

also studied.

6

Elsey et al. (1997) developed a dynamic model that predicts the variables affecting the

product quality in a single screw food extruder. The model was fitted to experimental data

and it was used in an inferential control algorithm. PI and MPC controllers were tested on

simulation level. MPC results came out to be much better than that of PI controller.

Nield et al. (2000) constructed a two input two output MIMO MPC scheme with constraints,

to control the product property (weight average molecular weight (Mw) and high molecular

weight end of molecular weight distribution) of LDPE in a reactive extrusion process by

manipulating the width and amplitude of square peroxide waves. The product property of

LDPE was inferred by using the measurements of an ‘inline wedge rheometer’. The controller

scheme was tested on actual plant. The results showed that, the control of consistency

index was good however power law index was not, due to the imposition of input

constraints, tuning of the controller for slow closed loop response and ill conditioning of the

system.

Haley and Mulvaney (2000) implemented a non-linear MPC on a food extruder in a cascade

manner. The objective was to obtain the product with the minimum piece density with the

constraint, where the other quality attributes should be acceptable. Responses of ‘specific

mechanical energy (SME)’ to screw speed and feed rate were modeled. Operating point set

points were obtained by an inferential model. In the cascade strategy, MPC was placed on

the inner loop and the cascade controller on the outer loop. Also a ratio controller was

utilized in a feed-forward manner to control the moisture content, and to eliminate the

disturbances in the feed. Furthermore the whole scheme was supplied with a feedback

algorithm on output, which corrects the model errors and disturbances. The results showed

that the scheme performed well for both disturbance rejection and set point tracking.

Wang and Tan (2000) developed a ‘dual-target predictive control strategy’, which could

track both the input and output set points in an optimal manner and applied to food

extruder. Die pressure and die temperature were the constructed model’s inputs and screw

speed, feed rate and moisture addition were the outputs. The case which the both input and

output set-points can be followed was called as ‘realizable case’ and the results showed that

the controller achieved its goal for this case.

Xiao et al. (2001) studied the control of coating properties of low-density polyethylene

(LDPE). The controlled and manipulated variables were the melt strength and the screw

7

speed respectively. The melt strength was measured by passing the extrudate through a

pulley system connected to a balance. The disturbance was introduced by changing the type

of LDPE fed, while the feed rate was kept unchanged. The PI and MPC were tuned offline

and their on-line performances were tested individually. In the work it was pointed that

there is a high order relation between the input and the output, which resulted in MPC’s

better performance.

Lee et al. (2002) tried to improve a previous work on the same subject where a blockage

phenomenon encountered in a food extruder. In order to prevent this phenomenon screw

speed was controlled by fuzzy reasoning. The fuzzy inputs were screw speed percentage and

torque percentage and the manipulated variables were water and feed flow and screw speed

and torque percentages. The study achieved its goal and the blockage was prevented.

Chen et al. (2003) proposed an ‘empirical viscosity model for quality control in the polymer

extrusion process’. In this work the viscosity was calculated using the parameters; screw

speed, melt temperature, geometric dimensions of the extruder and experimentally

determined material constants. The method was offered as an alternate and to overcome

the disadvantages of in-line capillary rheometer. The results of the work showed that the

proposed models can be applied to the product quality control using viscosity as the main

control parameter in the polymer extrusion process without implementing an in-line

rheometer, which would influence the output of the product.

Mudalamane and Bigio (2003) developed a first principles model and studied the effects of

external disturbances on the output fluctuations and transient behavior of extruders. Based

on the previous work, disturbances were categorized as high, medium and low frequency

disturbances. By using the model, damping effects of extruder design parameters on these

disturbances were studied. The simulation studies showed that, for a given screw design,

the process had a characteristic critical frequency above which the disturbances with higher

frequencies were damped out, however the ones with lower frequencies were not.

Choulak et al. (2004) developed a dynamic model for reactive extrusion in a twin screw

extruder, that predicted pressure, filling ratio, temperature and molar conversion. The model

was built as series of CSTRs that could be partially filled or fully filled with back flow.

Validation of the model was carried out by comparing the simulation data and the

experimental data. A good agreement between these data sets was observed. Aim of

8

automatic control of polymerization conditions in a twin screw extruder was mentioned but

no control studies were carried out.

Wang et al. (2004) presented a ‘three-stage approach to system identification in the

continuous time’. The three stages are; data acquisition using relay feedback, non-

parametric identification of the system step response, and parametric model fitting of the

identified step response. In the work discrete time noise model was integrated into

continuous time system identification. Experimental results were obtained by applying the

proposed method on a pilot scale food extruder and the results were presented in

comparison with model data.

Previdi et al. (2006) experimentally tested a prototype feedback control system for the

control of volumetric flow in a single screw extruder. The manipulated variables were the

barrel temperatures and die pressure. The work presents all the steps of the controller from

modeling to experimental testing. The results of the tests showed that the control scheme

responded well to disturbance rejections with small offsets on temperature and pressure.

2.2 PET Recycling and PET Degradation

Tanrattanakul et al. (1996) studied ‘toughening of PET by blending with a functionalized

polystyrene-poly(ethylene-co-butylene)-polystyrene (SEBS) block copolymer’. The aim was

to increase fracture strain that was affected by both blend composition and degradation

caused by process conditions. The processes utilized were extrusion and injection molding.

The intrinsic viscosity of PET was measured by Ubbelohde type viscometer using 60w/40w

phenol/tetracholoroethane as solvent. Results shoved that the blending increased the

fracture strain of PET. Torres et al. (2001) also made a similar study. They tried to improve

the thermal and mechanical properties of PET and recycled PET using chain extenders. They

also used the same solvent and equipment type for solution viscosity measurements.

Incarnato et al. (2000) aimed to increase the Mw of recycled PET to make it suitable for film

blowing and blow molding, by using pyromellitic dianhydrate (PDMA) as chain extender, in a

single step reactive extrusion. Effect of PDMA content on molecular structure was

investigated by using different concentrations of chain extenders, and then the extruded PET

samples were characterized. Results showed that a certain amount of PDMA increased Mw

and branching, broadened Mw/Mn, making recycled PET suitable for film blowing and blow

9

molding. Awaja and Davier (2004) made a similar study on an industrial scale extruder,

where PET was recycled by PDMA chain extender. In this work, effects of residence time and

temperature were also studied.

Pawlak et al. (2000) made a study on the ‘characterization of scrap PET’ in order to find

methods of characterization of recycled polymers and to show ‘general tendencies in

property change’. Applied techniques to achieve this goal were; differential scanning

calorimetry (DSC), Fourier transform infrared spectroscopy (FTIR), thermogravimetry (TGA),

dilute solution viscometry and dynamic viscosity measurement via capillary die. In the work,

it was pointed out that the main problem in polymer recycling was the segregation, which

was caused by the impurities that catalysis hydrolysis. In the study, scrap PET from

beverage bottles were obtained from different sources, and extruded in a laboratory scale

single screw extruder. Results showed that the presence of more than 50 ppm PVC made

PET unsuitable for more advanced processes such as film blowing.

Spinace et al. (2000) processed the PET used for production of soft drink bottles for five

times using SSE, and characterized rheological, mechanical and thermal properties of the

product including carboxylic end group number and melt flow index (MFI) analysis. The

study showed that after three processing cycles, changes in the crystallinity degree and in

the mechanical properties were occurred. It was pointed that the increase in MFI and

carboxylic end group concentration was a sign of mechanical degradation. The experiments

showed that the temperature profile changes were more affective using low screw speed,

which shows that the residence time has a direct affect on polymer degradation. Another

result of the study was that even after five processing cycles, thermal degradation behavior

of PET did not change.

Chelsea Center For Recycling And Economic Development (University of Massachusetts),

published the results of their laboratory study about “potential end uses for polyester fiber

waste”, on their technical report no.33 (2000). The PET fiber wastes were extruded using

different compositions of materials such as PET bottle waste, glass fiber and polycarbonate.

Products of these different compositions were tested separately with and without further

processing like molding. The effects of different factors, such as processing conditions,

presence of impurities and additive types, on the product’s quality and properties were

discussed.

10

Oromiehie and Mamizadeh (2004) studied PET beverage bottle recycling and methods to

improve its properties. The aim was to process and modify the mixture of virgin and recycled

grade PETs, with and without chain extender (PP-graft-MA) by different extrusion methods

and then to characterize the samples. Determination of tensile and thermal properties,

viscosity and Mw, and impact tests were carried out. Results showed that the intrinsic

viscosity ([η]) decreased as thermal process cycles and amount of recycled PET

concentration increased. Also, the chain extender improved the properties of the blends.

Assadi et al. (2004) studied the degradation types of PET during recycling by extrusion. The

experiments were performed using scraps of post-consumer PET, in a single screw extruder

at different temperatures. The Mw of the extruded samples were determined using steric

exclusion chrotomography (SEC), rheological tests and infrared measurements (IR). The

experiments were carried out using nitrogen and air environments with different air

pressures. A kinetic model for PET degradation was built, and results obtained from the

model were compared with the experimental ones. This comparison showed that model

results were in good match for nitrogen and oxygen environments.

Ajawa and Pawel (2005) made a brief review of PET recycling, taking the subject starting

from the synthesis of virgin PET, its properties, processing and applications. Use of chain

extenders was discussed, with the available machinery such as extruders to overcome the

molecular weight loss problem during recycling. Finally, to convert the RPET to a valuable

product, processes such as injection stretch blow molding (ISBM), which is a way to produce

PET bottles, were reviewed.

2.3 Model Predictive Control (MPC)

Marchetti et al. (1983) described the basics of a predictive control algorithm that was based

on discrete convolution models. Developed SISO predictive controller was compared to a

PID controller for three process models on simulation levels, and for an experimental

continuous stirred tank heater. Although predictive controller was superior on simulations, a

significant improvement was not observed in the experimental system. However, it was

pointed that the real advantage of predictive controllers was for MIMO cases.

Maurath et al. (1989) discussed the effects of the controller design parameters on closed

loop performance and robustness for an unconstrained SISO linear process. A stability

11

analysis that considers the plant/model mismatch was developed. Effects of design

parameters on controller’s performance were illustrated on several examples.

Brengel and Seider (1989) developed a model predictive algorithm for nonlinear MIMO case.

The control actions were calculated with a multi step predictor by linearizing ODEs. The

proposed algorithm was capable of easily handling of input and output constraints. In

simulation studies, the multi step predictor out performed the single step predictor.

Garcia et al. (1989) published a survey paper about MPC and compared several predictive

controller algorithms such as DMC, MAC and IMC. They pointed that the significant

advantage of MPC was the ‘flexible constraint handling capability’. Applications of MPC on

nonlinear systems were also investigated and concluded that the adjustment of MPC was

easier although it was not more robust than conventional feed-back controllers.

Morningred et al. (1992) developed an adaptive nonlinear controller similar to standard

linear model predictive controller. In the algorithm the number of tuning parameters could

be reduced to one. For the developed algorithm, effects of the modeling errors were shown,

and it was compared to PI, adaptive linear predictive controller and non-adaptive nonlinear

predictive controller, on a CSTR model. Results showed that the controller was

computationally efficient and could perform well even initially designed with modeling errors.

Meziou et al. (1996) used a dynamic CSTR model of an ethylene-propylene-diene

polymerization reactor to simulate the servo and regulatory performance of three input three

output MIMO DMC. Polynomial equations that relate the process’s gains to the magnitude of

the input change were derived, because the amplitude of the change in the input caused

variations in the gains of the process. Simulation results showed that the capability of MIMO

DMC to reduce the off-spec product amount caused by the set point changes and/or

disturbances.

Özkan et al. (2003) controlled the polymerization reaction in a CSTR with the developed MPC

algorithm that different linear models instead of a non-linear model. The objective function

included finite and infinite horizon cost components. The finite component made the system

move towards the desired operating point and the infinite component, having an upper

bound, brought the system to desired steady state operating point. Simulation results

showed that the proposed controller was successful at achieving the control goals.

12

Biagiola et al. (2005) published their case study about ‘use of state estimation for inferential

non-linear MPC’. The proposed non-linear estimator updates the state vector and estimates

the unmeasured disturbances where feed concentration was not measured. In the algorithm

concentration was inferred via temperature measurements. In simulation studies the

proposed non-linear observer non-linear controller structure is found to have good

performance to reject disturbances even in the presence of significant disturbance variations

and noisy measurements.

2.4 Inferential Control

Doyle III (1998) published a review in which inferential control, linear and non-linear

estimation techniques like moving horizon estimation methods or linearization by output

injection were presented and discussed. New theoretical approaches were presented on a

simple chemical reactor example.

Ogawa et al. (1999) built an inferential control scheme to control the melt index (MI) of the

product of a HDPE process. The MI was estimated by using the measurements of feed and

co-catalyst concentration and temperature measurements. The constructed inferential model

was based on a previous work of writers, simplifying it by means of computational burden.

The calculation of the control law was based on the relations of inferential model. The

system showed good regulatory and set point tracking responses on an industrial polyolefin

production.

Wang et al. (2001) implemented an inferential MPC algorithm on a food extruder, where

screw speed was manipulated variable and the bulk density of the product was controlled

variable, in order to control the product quality. The work also demonstrates the building a

continuous time dynamic model based on multi-rate sampled data. Experimental application

of the algorithm showed that the inferential control system maintains the product quality

within the specific ranges.

Bahar et al. (2004) utilized MPC to build an inferential control loop of an industrial multi

component batch distillation column. The MIMO MPC controlled the product compositions in

a feed-back manner, using the estimated values of the product compositions coming from

the artificial neural network (ANN) estimator. The estimator used the temperature

measurements from the selected trays. Simulation results showed that the unconstrained

13

and constrained MPC performances using ANN estimator, could be alternative to the

controller using direct composition values.

2.5 Singular Value Decomposition (SVD)

Klema and Laub (1980) have given a descriptive introduction to the singular value

decomposition (SVD) from the point of view of its computation and potential

applications. They emphasized certain important details of the implementation of

SVD on a digital computer. They also included a number of illustrative examples and

computed solutions, and concluded that singular value analysis forms a fundamental

basis of modern numerical linear algebra.

Rojas et al. (2004) proposed a strategy for the solution of quadratic performance index of

the optimal control law with constraints on inputs. A MIMO system whose Hessian of the

performance index had a large condition number was chosen for the illustration. Sub-optimal

control laws were obtained using SVD on Hessian matrix. Proposed strategy was compared

against MPC. Although the results were similar, proposed strategy held the advantage of

requiring no solution to the quadratic programming problem.

Zheng and Hoo (2004) used SVD technique to reduce the order of a distributed parameter

system (DPS). The system at hand was a tubular reactor, which was modeled as time series

model (infinite order). Order of this dynamic model was reduced to 3rd order in temperature

and 1st order in concentration. This linear model was then used as the plant model in a

quadratic dynamic model based controller (QDMC) and results were illustrated.

Luyben (2006) quantitatively compared the effectiveness of five different criteria for

selecting the temperature control trays in a distillation column. Their effectiveness were

tested on several systems ranging from ideal binary to azeotropic multi-component. Results

showed that among the tested criteria, SVD analysis provides a simple and effective method

for selecting tray locations.

14

CHAPTER III

BACKGROUND INFORMATION ON PET AND CONTROL TECHNIQUES

The detailed information about PET (synthesis, recycling and degradation), control

techniques (MPC and inferential control) and SVD analysis used in the study are given

below.

3.1 Poly(ethylene terephthalate) (PET)

PET is a thermoplastic resin of polyester family. It was patented in 1941 and was

commercially introduced as a textile fiber in 1953. The first PET bottle was patented in 1973.

PET has become a very important packaging material due to its good barrier and mechanical

properties. It makes a good barrier to gas, to alcohol (requires additional treatment) and to

many of the solvents. Furthermore semi-crystalline PET presents good thermal and

mechanical properties such as high melting temperature (approximately 250 oC).

PET can be synthesized by trans-esterification and/or condensation reactions as shown in

Figure 3.1. Depending on its processing conditions, amorphous (transparent) or semi-

crystalline (opaque and white) PET can be obtained.

The bulk synthesis of PET is carried out at 270 to 285 oC, with continuous removal of gas to

pressure below 1 mm Hg. The removal of methanol or water increases the molecular weight

of the polymer. If the methanol or the water were left in the same system, they would cause

a reverse reaction which would cause depolymerization.

15

The polymerization of PET can be carried out in the presence of a catalyst such as Sb, Ba,

Ca, Cd, Co, Pb, Mn, Mg, Ti, and Zn.

Figure 3.1: PET synthesis reactions: (a) trans-esterification reaction and (b) condensation

reaction [Ajawa and Pavel, 2005].

3.1.1 Recycling

PET recycling is the activity in which “post-consumer PET” or “recycled PET (RPET)”, mainly

formed of the collected beverage bottles, is reprocessed to a valuable product.

PET is non-degradable in nature, and its recycling is forced by environmental laws.

Furthermore, post-consumer PET is cheaper than virgin (non-processed) PET. For these

reasons PET recycling represents one of the most successful and widespread example of

polymer recycling. Today, approximately 1.5 million tons of PET is collected worldwide per

year. Petcore, the European trade association that fosters the collection and recycling of

PET, forecasts that in Europe alone, collection will exceed one million tons by 2010

[petcore.org].

16

During thermal recycling process, mechanical effects, moisture and presence of impurities

(PVC, adhesives, dyes, etc.), cause the loss of molecular weight (degradation) and lead to a

decrease in intrinsic viscosity ([η]), resulting in decrease in mechanical properties. RPET

having an intrinsic viscosity about 0.60 dl/g would be appropriate for fiber production, 0.65

dl/g for film production, 0.76 dl/g for bottle production and 0.85 dl/g for tire cord production

[Chelsea Center For Recycling And Economic Development, 2000].

RPET should satisfy the specifications given in Table 3.1 to be used as raw material.

Table 3.1: Minimum requirements for RPET flakes to be reprocessed [Ajawa and Pavel,

2005].

Property Value

Intrinsic viscosity ([η]) > 0.7 dl/g

Melting temperature (Tm) > 240 oC.

Water content < 0.02 wt.%

Flake size 0.4 mm < D < 8 mm

Dye content < 10 ppm

Yellowing index < 20

Metal content < 3 ppm

PVC content < 50 ppm

Polyolefin content < 10 ppm

3.1.2 Degradation

PET undergoes thermal, mechanical and hydrolytic chain scissions during recycling. Polymer

chains break by giving the volatile products mainly terephthalatic acid, acetaldehyde and

carbon monoxide. A sample reaction is given in Figure 3.2.

17

C O

O

H2C

H2C O C

O

C OH

O

+ H2C CH

O C

O

∆

Figure 3.2: Thermal degradation of PET [Karayannidis et al., 2000].

Mechanical degradation occurs due to the physical effects such as shear stress applied by

the extruder screws.

Hydrolytic degradation can be seen as the major effect reducing the molecular weight. This

type of chain scission is catalyzed by the impurities readily present in RPET, such as water

moisture, PVC, acid producing elements, dyes, etc. Acid alcohol condensation, catalyzed by

water is given in Figure 3.3 as an example. Hydrolytic degradation can be reduced by drying

the PET prior to processing.

Figure 3.3: Acid alcohol condensation of PET [Karayannidis and Psalida, 2000].

18

3.2 Control Techniques

In the control of the extruder system (see Chapter 4) MPC and PID techniques using

inferential models are used. A summary on MPC control technique and inferential control will

be given below.

3.2.1 MPC

Beginning from the late 1970’s predictive control techniques such as ‘Model Algorithmic

Control (MAC)’ [Richlet et al., 1978] (also known as Model Predictive Heuristic Control or

MPHC) or ‘Dynamic Matrix Control (DMC)’ [Cutler and Ramaker, 1980] began to gain

importance with the improving computer technology. Up to now, predictive control

techniques proved their efficiencies in many applications.

Although different MPC algorithms utilize different computation techniques, all utilize the

previous knowledge about plant dynamics (plant model) to predict what the plant output will

be after a definite time (prediction horizon), and calculates the next n number of control

actions (control horizon) in an optimal manner.

3.2.1.1 MPC Algorithm

Time

y

yM = aM

y1= a1

y2 = a2

y3 = a3

y4 = a4

h4

h2

h1

h3

4∆t 3∆t 2∆t ∆t 0

Steady state

M∆t

Figure 3.4: Open loop step response of a linear plant [Seborg et al., 1989].

19

In the MPC algorithm, future projection of the plant is calculated using the step response

coefficients (see Figure 3.4) in Equation 3.1 [Marchetti, 1981].

pTp ÊmAE +∆−=

−−

−−

−−

+

∆

∆

∆

−=

−

−

−

−+

+

++

++

++

Rn

R

n

n

Rn

n

nR

c

Rn

d

Rn

c

n

d

n

c

n

d

n

PE

PE

PE

m

m

m

a

a

a

aaa

yy

yy

yy

)1(

)1(

)1(

000

02

2

1

1

1

1

1

2

1

21

22

11

α

α

α

��

��

�

�

(3.1)

where subscript R denotes the prediction horizon and n denotes the sampling instant. The

superscript c denotes the corrected prediction and d denotes the desired value. The letters

y, m, a and E are used to represent the plant output, plant input, step response coefficients

and error respectively. The predicted errors, P, are calculated as follows.

Rlk

mhPl

k

N

ki

iknil

,....,2,1,

1 1

=

∆=∑ ∑

= +=

−+ (3.2)

If a perfect match between the predicted and the desired values is wanted (Ep = 0) then,

from Equation 3.1 the control action can simply calculated as,

pT ÊAm 1)(

−=∆ (3.3)

Equation 3.3 gives the control action at present sampling instant by predicting the next R

number of plant output. By applying the first element of ∆M vector and repeating the

procedure at every sampling instant the plant output is kept on desired values. But this

control law does not come out to be satisfactory as it tries to force the output to the desired

value at one sampling instant. To overcome this problem two proposed approaches are

Model Algorithmic Control (MAC) [Mehra et al., 1982; Richalet et al., 1978] and Dynamic

Matrix Control (DMC) [Cutler and Ramaker, 1980].

DMC reduces the dimension of ∆M from R (prediction horizon) to L (control horizon), and

only L number of future control actions are calculated. Thus, Equation 3.1 can be rewritten

as,

pp ÊMAE +∆−= (3.4)

20

A being the RxL “Dynamic Matrix” equal to the first L columns of AT. Optimal solution of

Equation 3.4 is obtained by minimizing the performance index by least squares. The solution

for the control action gives,

pTT ÊAAAM 1)(

−=∆ (3.5)

“One difficulty with the above control law is that if the ATA matrix is ill conditioned it can

result in large changes in the manipulated variable (ringing) or even unstable process”

[Marchetti, 1983]. This problem can be eliminated by introducing “weighting matrices” 1W

and 2W , which limit the manipulated variable moves, to the performance index.

pTPTp ÊWMEWEMJ 21)()( ∆+=∆ (3.6)

which results in the following control law:

pTT ÊWARAWAM 2

1

1 )(−+=∆ (3.7)

Here, again, the first control action is applied and new control law is calculated by observing

the plant output at each step. As only the first control action, nm∆ is applied, the control law

can be reduced to,

pT

nn mm ÊK+= −1 (3.8)

where elements of KT (gain matrix), are the elements of the first row of (ATA)-1AT in

Equation 3.5.

3.2.1.2 Constrained MPC

Up to this point no constraints are taken into account in the calculation of the control law.

However, in most processes, constraints should be imposed to the control actions, due to

the physical limitations and/or safety margins of the plant. Constraints may also be placed to

the plant output in order to prevent high deviations on product quality. When the constraints

are introduced, then the solution of the objective function becomes an optimization problem

in the following form:

21

maxmin

maxmin

max1min

21

:

)()(min

yyy

mmm

mmmm

tosubject

n

n

nn

pTPTp

≤≤

≤≤

∆≤−≤∆

∆+=∆

−

ÊWMEWEMJ

(3.9)

3.2.3 Inferential Control

In order to build a feed-back control algorithm, regardless of the controller type, on-line

measurement of the controlled output(s) is required. However, in quite a large number of

chemical process applications direct on-line measurement of the controlled output is late,

expensive or not available at all, which are the cases that limits the construction of the feed-

back control scheme [Seborg et al., 1989]. Feed-forward control could be utilized in such

cases being limited to the presence of measured disturbances and an appropriate model. For

the cases where neither on-line measurement of output nor unmeasured disturbances is

available, inferential control can be used to keep track of the unmeasured output. A block

diagram of inferential control loop is given in Figure 3.5.

Figure 3.5: Block diagram for inferential control loop.

22

3.3 Singular Value Decomposition (SVD)

SVD is an extension of singular value analysis (SVA). It is used to determine the rank and

condition of a matrix and also to determine the strengths and weaknesses of a set of

equations [wikipedia.org]. In control point of view, SVD is utilized for controlled-manipulated

variable pairing for a MIMO system. In the frame of this work, only control aspect of SVD

will be given.

In the simplest form, SVD is the factorization of a rectangular matrix, as follows:

TVUK Σ= (3.10)

Where K is the mxn matrix, U is the nxm orthonormal matrix called left singular matrix

that contains output basis vector directions forK ,Σ is an nxm diagonal matrix of singular

values that can be thought as scalar gains, and V is an mxm orthonormal matrix called

right singular matrix that contains input basis vector directions for K .

Steady state relations of a MIMO system, with n number of outputs and m number of inputs,

can be expressed in vector-matrix form as:

GMY = (3.11)

Where, Y is the output vector, G is the steady state gain matrix and M is the input vector.

It is possible to find which output is sensitive to which input by applying SVD toG . From the

resulting U andV matrices, largest element of 1st column of U ( ny∆ ) is most sensitive to

the changes in the largest element of 1st column of V ( mm ), largest element of 2nd column

of U to the largest element of 2nd column ofV , etc.

For the cases where number of inputs and outputs of the system are not equal to each

other, the pairings corresponding to the zero elements of Σ does not have to be calculated.

Such a calculation is called as the compact SVD.

Another aspect of SVD is the Condition Number, CN. It is defined as the ratio of the largest

and the smallest non-zero singular values:

23

r

lCNσ

σ= (3.12)

For a large CN of G, the system is said to be ill-conditioned. Furthermore, if G is singular,

then it is ill-conditioned.

24

CHAPTER IV

EXPERIMENTAL STUDIES

In the experimental studies done, RPET is extruded at different processing conditions and

samples are collected for molecular weight determination to obtain degradation data.

Materials used, experimental procedure, setup and machinery are presented in this chapter.

4.1 Properties of RPET and Trifluoroacetic Acid (TFA)

RPET is used in the form of flakes in the experiments. The properties of the RPET as

specified by the supplier (AdvanSA, Adana) are presented in Table 4.1.

Two commonly used solvents for PET are, 40wt% tetracholoro ethane – 60wt% phenol

mixture and trifluoroacetic acid (TFA). Being carcinogenic, the first one is eliminated and

TFA is used as the solvent. Figure 4.1 shows the molecular structure of TFA.

Figure 4.1: Molecular formula of trifluoroacetic acid (TFA).

Mark Hauwing constants for PET-TFA solution at 25 oC are 41014

−= xK and 65.0=α

[Brandrub and Immergut, 1989]:

25

Table 4.1: Properties of RPET resin (AdvanSA)

PVC 60

Polyethylene 5

Metal pieces 0

Adhesive 10

Contaminants (ppm)

Paper pieces 3

Value L, Shining 66.1

Value B, Yellowness 2.6 Lighting Characteristics

Value A, Redness -2.0

Intrinsic Viscosity ([η]) 0.750 dl/g

Glass Transition Temperature (Tg) 60 ºC Material Properties

Melting Temperature (Tm) 255 ºC – 260 ºC

4.2 Experimental Setup

RPET is extruded using a laboratory scale co-rotating twin screw extruder (Thermoprism TSE

16TC, L/D = 24) as shown in Figure 4.2 to obtain degradation data. The schematic drawing

for the extruder system is given in Figure 4.3.

The extruder had five electrical heaters through the barrel, whose temperatures can be set

separately. The cooling is provided by passing through cooling water in the barrel. The feed

is supplied via a brabender type feeder whose screw speed can be adjusted. The parameters

that can be set from the control panel of the extruder are screw speed, feed rate (feeder

screw speed), and temperatures of each 5 heating zones.

The available measurements from the control panel are screw speed, temperatures of each

five heating zones, melt temperatures from four distinct points, die pressure and

temperature. A photograph of the control panel is given in Figure 4.4.

26

Figure 4.2: Extruder used for experiments.

Figure 4.3: Schematic drawing for the experimental setup.

27

Figure 4.4: Control panel of the extruder (a: die temperature and pressure, b: melt

temperatures, c: barrel temperatures, d: screw speed and torque, e: main feed rate).

4.3 Experimental Procedure

The experiments are carried out in two phases. In the first phase preliminary and steady

state experiments are done and in the second phase dynamic experiments are carried out.

4.3.1 Preliminary Experiments

In the preliminary experiments the effect of temperature on viscosity and molecular weight

is investigated. Also the calibration of the extruder is done.

Effect of Temperature: The effect of temperature on RPET degradation is studied by

eliminating other parameters. Samples are packed firmly in aluminum foil and are held in oil

bath at 4 different temperatures (270, 280, 290, 300oC) for 1, 3 and 5 minutes. The intrinsic

viscosities ([η]) of these 12 samples are determined by dilute solution viscometry (see

Appendix A).

28

The choice of studied temperature and time ranges are based on the studied temperature

and residence time range of extrusion. The residence time range of the extrusion depends

upon the screw speed in a twin screw extruder (see Appendix B). Thus in these

experiments, effect of screw speed in means of residence time is also included indirectly.

Auto ignition temperature of silicone oil limited the testing of higher temperatures.

Calibration of the Extruder: In order to find the flow rate (g/min) equivalent of feed rate

setting of extruder control panel, the feed flow is collected for one minute intervals and

weighted. Different feed rate settings such as 25, 50, 75 and 100, are used and average

values for flow rates are found by repeating the experiments.

The residence times for the studied screw speeds (50, 125, 200, 275, 350, 425 and 500

rpm) are measured by using carbon-black containing polyethylene (PE) pellets as indicator.

The indicator pellets are dropped manually into the feeding point while the main feed

(RPET) is being fed. The time when the product color (originally semi-transparent or opaque

white) changed from gray to black is recorded as the average residence time (ART).

4.3.2 Steady State Experiments

The aim of these experiments is to determine the intrinsic viscosities ([η]) and

corresponding molecular weights (Mv) of products processed at different operating

conditions.

RPET is extruded at 3 different temperatures1, 270, 290 and 310 oC; 4 different feed rate

settings, 25, 50, 75 and 100; and 7 different screw speeds, 50, 125, 200, 275, 350, 425 and

500 rpm, resulting in 84 samples.

After the system is reached its steady state operating conditions (t >> ART) (see Appendix

B) 25-30 grams of samples are collected from the extruded product. The samples are too

1 The barrel temperatures’ setting is a parameter itself for output properties. In this study, the temperature of each

zone is set constant and equal to each other, to eliminate the effect of this parameter. For example, T = 270 oC

means that all the temperatures along the barrel are set to 270 oC.

29

thin and elastic to be pelletized in the pelletizer, thus they are cut into small pieces at hand

to homogenize. From homogenized sample particles, 0.06 gram is weighed for intrinsic

viscosity ([η]) measurements.

4.3.3 Dynamic Experiments

These experiments are aimed to collect the necessary data to model the dynamic response

of the system output (Mv) to the changes in the process parameters (SS, FR and T). In other

words aim is to find out the pattern that Mv follows from one operating condition to other.

Using the data obtained from steady state experiments, operating conditions are selected as

SS = 100 rpm, FR = 7.12 g/min (feed rate setting = 75) and T = 270 oC, for a product

having an intrinsic viscosity value of [η] = 0.6 dl/g, which is suitable for fiber production

[Chelsea Center For Recycling And Economic Development, 2000].

Thus, step changes given in Table 4.2 are introduced one at a time starting from this

operating point. Each step change is given after bringing the system to the initial steady-

state operating point.

Table 4.2: Step changes given to the process variables.

Variable

Initial steady state

value

Plus

change

Minus

change

Screw speed 100 rpm 25 rpm 50 rpm

Feed rate1 7.12 g/gmol 1.04 g/gmol 1.42 g/gmol

Temperature 270 oC 20 oC 20 oC

For screw speed and feed rate changes, the samples are collected at 10 second intervals,

whereas samples are collected at 20 second intervals for temperature changes. This is

because the system responded to the changes in temperature more slowly. The first sample

1 It should be noted that, for step changes on feed rate, feed rate setting of the control panel are taken as the

basis. Calibration data for the feed rate setting and the material flow rate can be found in Appendix C

30

(sample at t = 0) is collected at the time when the step change is introduced. For every

sampling time, outcoming product is collected for 5 seconds. Then, these samples are

broken into small pieces manually for homogenization and 0.06 gram of each sample is

weighted for intrinsic viscosity measurement.

31

CHAPTER V

RESULTS AND DISCUSSIONS

The results of preliminary experiments for the verification of supplier data for RPET

specifications and the experiments on the effect of temperature on degradation of PET are

given below. These will be followed by the results of experiments on the extruder under

steady-state and transient conditions. The results of simulation and modeling studies will be

introduced with discussions.

5.1 Preliminary Experimental Results

Verification of Supplier’s Data: The intrinsic viscosity ([η]) and viscosity average molecular

weight (Mv) of RPET is measured without processing the RPET samples, in order to check

with the supplier’s data. The results of these experiments are given in Table 5.1.

It is found that the experimentally measured value for [η] = 0.75 and corresponding

molecular weight value, Mv = 18363 g/gmol are in a very good agreement with the

supplier’s data with an error of + 0.001 dl/g in [η] and + 26 g/gmol in Mv.

Effect of Temperature on RPET Degradation: As given in Chapter 4, these experiments are

performed in oil bath where samples of RPET are exposed to temperature in the range of

270 - 300 oC for different time durations. The results in terms of Mv are given in Figure 5.1.

32

Table 5.1: Measured intrinsic viscosity and molecular weight of unprocessed RPET.

Sample No.

Intrinsic viscosity ([η])

(dl/g)

Viscosity average

molecular weight (Mv)

(g/gmol)

1 0.750 18363

2 0.747 18248

3 0.751 18401

Avarage 0.749 18337

Supplier Data 0.750 18363

Figure 5.1: The effect of temperature on molecular weight of RPET.

Figure 5.1 shows that Mv does not follow a simple decreasing trend with increasing

temperature with time. Instead, there exist some optimal points (or regions) where Mv

increases and then decreases. Although not studied in this work, one possible cause may be

the branching or crosslinking, which was also observed in the previous studies [Spinace et

al., 2000; Pawlak et al., 2000; Assadi et al., 2004] on RPET degradation. It is known that

heat causes the longer polymer chains break into smaller ones. In case of branching or

crosslinking, these smaller chains form new bonds on the backbone of another chain, as a

result of coupling alkyl radicals generated by the oxidation chain process, with the effect of

33

temperature [Assadi et al., 2004]. Furthermore, the contaminants of RPET, such as dyes and

adhesives may have a catalyzing effect on this reaction.

Thus, it can be seen from Figure 5.1, Mv decrease within for temperatures 270, 290 and 300 oC. The only disagreement with this observation is the trend for 280 oC.

5.2 Steady State Experimental Results

Steady-state experiments with the laboratory scale co-rotating twin screw extruder are done

in order to find the operating point which will produce an output with the desired intrinsic

viscosity and molecular weight values of [η] = 0.6 dl/g and Mv = 11500 g/mol [Chelsea

Center For Recycling And Economic Development, 2000]. A product with these values is

suitable for the fiber production, one of the most important areas of use of the RPET.

Thus, RPET is extruded at three different barrel temperatures (270, 290 and 310oC), four

different feed rates (3.85, 5.70, 7.12 and 8.16 g/min), and seven different screw speeds

(50, 125, 200, 275, 350, 425 and 500 rpm). The viscosities of the samples from these 84

runs are measured to obtain the Mv relations as a function of process parameters. The

results are given in Figures 5.2 to 5.4.

Figures 5.2 to 5.4 indicate that molecular weights of samples (see Appendix C) are as a

function of screw speed at different temperatures and feed rates. The range of molecular

weight of samples are approximately 2000 - 15000 for 270 oC, 6000 - 13000 for 290 oC, and

2000 - 11000 for 310 oC. These trends prove the degradation of Mv with temperature.

A similar generalization cannot be done for the effect of screw speed and feed rate on Mv.

Shear stress exerted by the screws and the filling ratio, which is a function of both the screw

speed and the feed rate are the two most probable effects on Mv degradation. The effect of

filling ratio can be explained with the heat and shear consumed by unit amount of material

in the extruder. Thus, oscillatory behavior of molecular weight can be due to the adverse

effects of these two variables. Increasing the screw speed increases the shear stress applied

and decreases the filling ratio, ending up in a decrease in Mv.

34

Figure 5.2: Effect of SS and FR on Mv at T = 270 oC.


35


After the steady state measurements are completed, intrinsic viscosities of randomly

selected nine samples are measured again to check the consistency of the previous

measurements. As can be seen from Table 5.2, the precision of the measurements are

acceptable.

In order to produce a product with a Mv of 11500 g/mol, it can be concluded from Figures

5.2 to 5.4 (◊ marked points) that, this value can be obtained by the parameters SS = 100

rpm, FR = 7.12 g/min and T = 270 oC. It is important to note that the operating point must

be such that possible variations in the variables must not result in degradation in Mv. This is

a required flexibility for a safe operation. If, for example, process temperature is chosen as

290 oC, then Mv for the product can only be achieved in a very restricted range. Due to large

oscillation in the behavior of Mv as a function of SS and FR, at T = 310 oC Mv = 11500

g/gmol cannot be obtained for any of the SS and FR values in the selected ranges.

36

Table 5.2: Results of consistency check

Previous measurement Consistency measurement

Sample from:

[η]

(dl/g)

Mv

(g/gmol)

[η]

(dl/g)

Mv

(g/gmol)

T: 270 oC, FR: 25, SS: 200 rpm 0.526 10549 0.523 10221

T: 270 oC, FR: 75, SS: 125 rpm 0.632 14053 0.634 14123

T: 270 oC, FR: 100, SS: 500 rpm 0.570 11966 0.563 11731

T: 290 oC, FR: 25, SS: 50 rpm 0.393 6689 0.392 6657

T: 290 oC, FR: 50, SS: 425 rpm 0.614 13344 0.614 13426

T: 290 oC, FR: 100, SS: 500 rpm 0.550 11310 0.558 11565

T: 310 oC, FR: 25, SS: 200 rpm 0.350 5584 0.347 5497

T: 310 oC, FR: 25, SS: 200 rpm 0.479 9116 0.483 9241

T: 310 oC, FR: 25, SS: 200 rpm 0.484 9250 0.486 9307

5.3 Dynamic Experimental Results

The dynamic experiments which are aimed to find dynamic models for the process under the

change of the manipulated variables and in the presence of disturbances are done with the

changes of screw speed from 100 rpm to 125 rpm and from 100 rpm to 50 rpm, feed rate

setting from 75 (7.12 g/min) to 100 (8.16 g/min) and from 75 (7.12 g/min) to 50 (5.70

g/min), barrel temperatures from 270 oC to 290 oC and from 270 oC to 250 oC. Each change

is introduced while keeping the other parameters constant at operating point.

The step sizes are chosen to keep the changes in the variables in the range of steady-state

experiments. The results of dynamic experiments are given in Figures 5.5 to 5.7 for positive

and negative step changes in screw speed, feed rate and temperature respectively.

It is observed from the Figures 5.5 to 5.7 that, the system responds differently for different

inputs proving the highly non-linear structure of the process. In Figure 5.5.a response of the

system to a positive deviation in screw speed is given while in Figure 5.5.b response in Mv to

a negative deviation in SS is given. The natures of Figure 5.5.a and Figure 5.5.b are

completely different. This trend is also seen in Figure 5.6 and 5.7. Accordingly, it is decided

to use different process models depending upon the input deviations or simply depending

upon the deviation in the output, Mv.

37

Figure 5.5: System responses to (a) positive (+25 rpm) and (b) negative (-50 rpm) steps

on screw speed.

Figure 5.5 shows that screw speed or simply average residence time (ART) for the RPET in

the extruder (see Table B.1) is more dominant on Mv than other mechanical effects in terms

of degradation.

Considering Figure 5.6, as feed rate increases, amount of material in the extruder (filling

ratio) increases, therefore heat and shear consumed per unit amount of material decreases,

resulting in a increase in Mv of the product. Thus, as feed rate increases (Figure 5.6.a) Mv

will increase (less degradation occurs), and as feed rate decreases (Figure 5.6.b) Mv will

decrease (more degradation occurs).

In Figure 5.7 effect of temperature on Mv degradation is shown at constant screw speed and

feed rate. The explanation of this trend cannot be done with the results of steady-state

experiments and with the known literature, except that the formation of new bonds

(branching/crosslinking) is lower (or does not exist at all) at 250 oC compared to 290 oC.

38

Figure 5.6: System responses to (a) positive (+1.04 g/min) (+25) and (b) negative (-1.42

g/min) (-25) steps on feed rate.

Figure 5.7: System responses to (a) positive (+20 oC) and (b) negative (-20 oC) steps on

temperature.

39

5.4 Modeling Studies

As stated before, the experimental data obtained from the dynamic experiments are aimed

to construct dynamic models of the process for control purposes. The experimental results

which are illustrated in Figures 5.5 to 5.7 are revealed that the system behavior is highly

nonlinear in nature. Therefore, the process needed to be modeled separately for the

different changes in inputs as positive or negative. Thus, two separate models, one

representing the response to positive and the other one to negative changes are obtained

for screw speed, feed rate and temperature inputs separately and named as ‘SS Model’, ‘FR

Model’ and ‘T Model’. This model representation is shown in Figure 5.8.

As can be seen from the Figure 5.8, the input of each dynamic model is a different input

variable of the extruder and the output of the model is the molecular weight of the product

representing the dynamic behavior of the extruder machinery as a whole.

The experimental data must be represented analytically by a suitable model in order to be

used in the control algorithm. Thus, in the modeling studies three different modeling

techniques (single transfer functions, dual transfer functions and convolution models) are

tried in order to obtain the best fit. In the first technique a transfer function is used for

input-output relationship for the extrusion. In the second technique, two different transfer

functions are used for two different consecutive periods throughout the transient response

of the system. In the third technique, discrete convolution model is used. MATLAB software

is utilized in all these studies.

In the modeling studies, experimental data are normalized with respect to initial steady state

operating point. In normalization, each deviation variable (difference from the initial steady-

state value) is divided by the magnitude of the step input. Consequently, it is assumed that

the process is behaving linearly in the inputs’ magnitude ranges. This assumption will be

verified by the simulation studies done for different magnitudes of the inputs.

40

Figure 5.8: Model representation of the system.

The formulation of the normalization can be expressed as:

m

MMM

vnvN

v∆

−=∆ 0 (5.1)

where N

vM∆ is the normalized molecular weight of the product, m is the magnitude of the

input (SS, FR or T) and n is the data points.

41

5.4.1 Modeling Technique 1: Single Transfer Functions

In this method, the process is tried to be modeled by a single transfer function. Different

forms of transfer functions tested by changing the order (1st, 2nd and 3rd) and by changing

the nature (integrator, dead-time and zero-poles).

‘System identification toolbox’ of MATLAB is used to identify the system automatically.

However, toolbox failed in identification and custom codes (see Appendix D) are generated.

In modeling studies, for every parameter of the transfer function utilized a step response of

the transfer function is obtained for a specific input and the absolute error, between it and

the experimental data points are evaluated. The best fits giving the minimum integral

absolute error (IAE) scores are given in Table 5.3 with their normalized IAE scores (divided

by steady-state gains). The response curves to unit step changes are given in Figures 5.9 to

5.11 in terms of normalized molecular weight ( N

vM∆ ) as a function of time.

Figure 5.9: Plant and model responses to a (a) positive and (b) negative unit step

changes in SS (single transfer function case).

42

Table 5.3: Single transfer function models and their normalized IAE scores.

43


changes in FR (single transfer function case).


changes in T (single transfer function case).

44

As can be seen from Figures 5.9 to 5.11 the system dynamics can be modeled by a single

transfer function for each input. However the errors between the experimental data points

and the models are large (large IAE scores) especially for the positive input in SS and

negative input in FR.

5.4.2 Modeling Technique 2: Dual Transfer Functions

In the second method, two transfer functions are used to model the system. As stated

before, the system under study is highly non-linear (Figures 5.5 to 5.7). Although transient

positive and negative responses are considered differently, in modeling, also they can be

modeled using different transfer functions for the whole time domain for a single input.

Thus, for each of the six experimental data, two transfer functions can be used in such a

way that, at the point where first transfer function looses the track of the experimental data,

second one activates by deactivating the first one. Parameters of these transfer functions

are adjusted by trial and error procedure as in the case of single transfer function modeling.

In Figures 5.12 to 5.14 the response curves to unit step inputs are given using this 2nd

modeling technique and in Table 5.4 the transfer function models, their acting time domains

and normalized IAE scores (divided by steady-state gains) are given.

Table 5.4: Dual transfer function models and their normalized IAE scores.

Model TF1 TF2 Switch Time IAE Scores

SS+

1222486

4.10245932 ++

+−

ss

s

11155

12.74582 ++

+

ss

s

120th sec. 0.25

SS- s

ess

9

21762100

1.192 −

++

1554

4.62 ++ ss

200th sec. 0.80

FR+

1483008

8.472 ++ ss

124245

12.22 ++

−

ss

130th sec. 0.34

FR- s

ess

5

211685903

64.341 −

++

1979

36.2102 ++

−

ss

110th sec. 0.38

T+

122843

4.662 ++ ss

1251262

55.232 ++

−

ss

120th sec 0.39

T-

1284818

4.1602 ++ ss

125287

7.6828162 ++

+−

ss

s

180th sec. 0.45

45


changes in SS (dual transfer functions case).


changes in FR (dual transfer functions case).

46


changes in T (dual transfer functions case).

As can be seen from the Figures 5.12 to 5.14 and considering the IAE scores from Table 5.4,

these models fit the experimental data much better than single transfer functions. However,

during the control operation, manipulated variable (SS) changes and the controlled variable

(Mv) is tried to be kept constant or to track the set point. In these manipulations the model

based controller has to be switched on different models (positive/negative/TF1/TF2). This

cannot be solved by a single algorithm. Therefore, this modeling technique is not considered

to be user friendly.

5.4.3 Modeling Technique 3: Discrete Convolution Models

It is well known that in the model predictive control, MPC, algorithm discrete convolution

models are widely and effectively used. In the convolution model, the output is evaluated by

using the experimentally obtained step response coefficients (see Figure 3.4) and the past

values of the manipulated variables. The model equation is given in Equation 5.2 [Seborg et

al., 1989]. Thus, the model is considered to represent linearly the responses to all different

inputs in the range studied.

47

inn

n

in may −+

=+ ∆∑=∆ 1

11 (5.2)

where y∆ is the output in deviation form, a is the step response coefficient and m∆ is the

manipulated variable (model input) in deviation form.

In this study the experimentally obtained step response data can be curve fitted as ‘shape

preserving function’ using ‘curve fit toolbox’ of MATLAB and step response coefficients (an)

are obtained as a function of time. Comparison of model and actual system outputs are

given in Figures 5.15 to 5.17.


changes in SS (discrete convolution models case).

48


changes in FR (discrete convolution models case).


changes in T (discrete convolution models case).

49

In Figures 5.15 to 5.17, the digital model outputs (in 1 second intervals) are shown in

comparison to experimental data given in 10 seconds intervals. The disadvantage using this

method is that, they are not compatible with MATLAB, i.e. there does not exist a pre-

constructed library, function or toolbox for the discrete convolution models. Thus, codes

working in continuous time have to be written for this case (see Appendix D).

5.5 Control Studies

After the dynamic behavior of the plant is modeled, the final step is to design a proper

controller in a control loop. In the design of the control loop, the first step is the choice of

manipulated variable(s) for the control of system output. After deciding on manipulated and

controlled variable pair, a control scheme is designed and on this scheme, MPC and PID

controllers are tested and compared. The system under study (Figure 5.8) is a multi input

single output (MISO) system. The inputs are SS, FR and T, and the output is the product

property. The output, which is the product property, is determined in terms of molecular

weight (Mv) and this is measured by intrinsic viscosity ([η]).

5.5.1 Singular Value Decomposition

The design of a control system, when there are more than one input, necessitates the use of

the singular value decomposition, SVD, technique for the selection of the manipulated –

controlled variable pairs. By this method, it is possible to find out which output is most

sensitive to which input. SVD analysis calculations for the system under study are given

below.

As stated before, the system is highly non-linear and can only be represented by different

models (positive and negative) for different inputs (as positive and negative).

The steady-state Mv can be expressed in terms of input variables for positive and negative

models as given in Equations 5.3.a and 5.3.b respectively.

[ ]

=+

+

+

++++

T

FR

SS

KKKM TFRSSv

(5.3.a)

50

[ ]

=−

−

−

−−−−

T

FR

SS

KKKM TFRSSv

(5.3.b)

where K+ and K- symbols denote the steady state gains for the corresponding inputs, and

superscripts ‘+’ and ‘-’ denotes the relevant model parts. Replacing steady state gain values

in Equations 5.3.a and 5.3.b gives:

[ ]

=+

+

+

+

T

FR

SS

M v 4.668.474.102

[ ]

=−

−

−

−

T

FR

SS

M v 4.1606.3611.192

(5.4.a)

(5.4.b)

Decomposing steady state gain matrices (K+ and K-) in Equation 5.4.a and 5.4.b gives:

[ ][ ]

=+

0.9253 0.1037- 0.3647

0.1037- 0.8559 0.5066

0.3647- 0.5066- 0.7813

000708.1311K

[ ][ ]

=−

0.5294 0.2087- 0.8223

0.2087- 0.9074 0.3647

0.8223- 0.3647- 0.4368

00439.78871K

(5.5.a)

(5.5.b)

SVD1 shows that the best pair is screw speed – molecular weight for positive model and

temperature – molecular weight for negative model, considering the largest elements in the

first columns of VT matrices in the Equations 5.5.a and 5.5.b.

However, in control it is not possible to change the manipulated variable when the input

changes. Thus, beside SVD recommendation, choice of the variables must also be based on

expert knowledge of the process. Among the inputs, manipulating the screw speed is the

fastest and the easiest. The extruder responds to the changes in the screw speed setting

1 Second and the third column of the VT matrices should be ommitted as they correspond to the zero elements of

the singular value vector (Σ ).

51

almost instantaneously. On the other hand, the changes in the temperature are responded

much more slowly. Furthermore, in an actual plant, the feed flow rate is kept constant for

the sake of the operations following the extrusion. For these reasons, screw speed is chosen

as the manipulated variable, while temperature and feed rate are considered to be

disturbances.

The choice of the disturbances is based on the facts that a blockage or disorder in the feed

flow is always possible. Also, there can be failures in the heaters or changes in cooling

water, resulting in unwanted changes in the barrel temperature.

It should be noted that no measurement of input FR is available in the process other than

the adjustment of the extruder. But having the model, disturbances on the feed flow is also

tested as if measurements are available.

5.5.2 Design of Control Scheme

As stated before, the objective of the control system is to control the molecular weight (Mv)

of the product leaving the extruder under the effect of disturbances or in the case of set

point changes. However, as in this study, online measurement of the desired property (Mv)

may not be possible. Thus, a modified inferential control scheme can be used, in which an

estimator estimates the desired property by using the secondary online measurements

available from the plant and supplies necessary feedback data. The product quality can be

determined by on-line viscosity measurements which can give the Mv of the product. Thus, a

feedback control loop is designed as shown in Figure 5.18. In this scheme among the input

variables such as SS, FR and T; SS is chosen as the manipulated variable (see Chapter

5.5.1) while FR and T are chosen as probable disturbances. Other disturbances to the

system may be the PET composition. This, of course, can be measured from the batch

introduced and this information can be used for first settings of the operation for a required

product quality. As it is found experimentally that the system is highly non-linear, the

controllers are made of two compartments, +cG and −

cG . +cG is using of the ‘positive’ process

model and the other, −cG , is using of the ‘negative’ process model. A ‘switch’ in the entrance

of the controller decides which model to run, basing on the deviation of Mv from its initial

desired value.

52

Figure 5.18: Designed control system.

Also, both in the disturbance models +pG and −

pG , and in the estimator +EG and −

EG exist.

5.5.3 PID Control

PID controllers are used in the designed control system. PID controllers are designed using

the convolution models of the system by utilizing ‘continuous cycling’ or ‘Ziegler-Nichols’

method with ‘modified Z-N settings’. PID parameters are calculated by using the

relationships given in the literature [Seborg et al., 1989] (see Appendix E), and the results

are given in Table 5.5. In Figure 5.19 responses of set point tracking for the PID controllers

are given. The set points are the 10% of the model gains. The normalized IAE scores for

positive and negative models are given in Table 5.5. IAE scores are normalized with respect

to steady-state gains and time ranges.

Table 5.5: PID settings utilizing modified Z-N method (Some Overshoot) [Seborg et al.,

1989], and normalized IAE scores of set point tracking (Figure 5.19).

cK iτ Dτ IAE Score

PID+ 0.0025 2.7399x105 0.1513 4.85

PID- 0.0058 1.0357x104 0.2165 0.01

53

Figure 5.19: Set point tracking of PID controllers of (a) positive and (b) negative models,

tuned using modified Z-N settings (Some Overshoot) [Seborg et al., 1989].

As can be seen from the Figure 5.19 the system shows an unstable performance for both

positive and negative inputs and settling times are very large for these settings. Therefore,

the controllers are further fine tuned by trial – error procedure. The fine tuned PID settings

are given in Table 5.6.

Table 5.6: Fine tuned PID settings for PID+ and PID-.

cK iτ Dτ

PID+ 2.50x10-3 6.85x10-5 3.03x10-2

PID- 3.50x10-3 6.21x10-5 2.60x10-2

These PID settings are very small and cannot be implemented on a nominal operating

industrial plant controller. However, if a computer is used for the PID controller, then these

settings can be implemented. Therefore, in this study, for comparison with the MPC, which

can only be implemented via computers, the PID settings as given in Table 5.6 are used.

Furthermore, constraints are also placed on PID outputs (SS) to be able to compare PID

54

controller with the MPC. The lower limit implemented on PID+ is 0 rpm and the upper and

lower limits for PID- are 0 rpm and -50 rpm respectively.

In order to observe the set point tracking response of the controller, step changes listed in

Table 5.7 are introduced. For the disturbance studies, the disturbances are introduced as

+25 in FR (corresponding to +1.04 g/gmol and -1.42 g/gmol), and +10 oC in T.

Table 5.7: The magnitudes and the time of changes of the given set point changes.

Time (s) 0 1000 2000 3000 4000 5000

Magnitude (g/mol) +1200

(+10.4%)

-550

(-4.8%)

-650

(-5.7%)

-9050

(-78.7%)

+7450

(+64.8%)

+1600

(+13.9%)

Control loop is built using ‘SIMULINK toolbox’ of MATLAB for the PID controller (see

Appendix D). PID performances are tested for set point tracking and disturbance rejection

cases. The results are given in Figures 5.21 to 5.29.

5.5.4 Model Predictive Control (MPC)

In the design and testing of MPC, as for PID controllers, two parallel working SISO MPCs are

constructed using the ‘model predictive control toolbox’ of MATLAB, for non-linear

constrained MPC.

5.5.4.1 Process Models for MPC

In order to generate a step response process model that can be used in MPC algorithm, ARX

models are formulated and used to obtain the molecular weight response to a step change

in screw speed. This is shown in Figure 5.20 for the positive and the negative inputs. As can

be seen from the Figures 5.5 to 5.7, the system is highly non-linear and the response

behavior is very different for input values of positive and negative (Figure 5.20.a and

5.20.b).

55

Figure 5.20: Responses of ARX models to (a) positive and (b) negative unit step changes

in SS.

5.5.4.2 MPC Tuning

Tuning of predictive controllers are carried out by trial and error procedure, basing on the

recommendations in the literature [Seborg et al., 1989]. MPC tuning parameters are given in

Table 5.8.

In the preliminary control studies it is seen that MPC fails to reject the disturbances. This

problem was also seen in a previous study [Obut, 2005]. MPC algorithm assumes that the

disturbances have a ‘constant-linear’ nature. However, in this work, the trends of

disturbances resemble the response of a first order transfer function to a step input. Thus,

designed MPCs usually fail to handle these disturbances. This problem is solved by changing

the disturbance estimation type from constant to ‘step’ in +cG and to ‘ramp’ in −

cG .

56

Table 5.8: MPC tuning parameters.

Parameter MPC+ MPC-

Prediction Horizon 200 100

Control Horizon 1 1

Output weights / Input weights 1 1

Disturbance Type And

Magnitude Step (20) Ramp (10)

Constraints On Manipulated

Variables

Min: 0

Max: ∞

Min:-50

Max:0

Designed MPCs are tested for set point tracking (see Table 5.7) and disturbance rejection

performances. In Figure 5.21 the MPC responses for set point tracking is given together with

PID performances for comparison.

Figure 5.21: Set point tracking responses of PID and MPC.

Normalized IAE scores calculated from Figure 5.21 are given in Table 5.9. In normalization

of IAE scores, calculated IAE score is divided by the input magnitude and by the time range

in which it is calculated.

57

Table 5.9: Normalized IAE scores of controllers throughout set point tracking.

Time range (s) 0 to

1000

1000 to

2000

2000 to

3000

3000 to

4000

4000 to

5000

5000 to

6000

PID 1.00 1.19 0.01 0.99 0.21 0.00

MPC 0.99 1.17 0.01 0.99 0.21 0.00

The performances of the controllers for tracking the set point for the given step sizes are

very good and very similar to each other. Although in Figure 5.21 the deviations for larger

step sizes (3000<t<4000) are larger, this cannot be reflected in normalized IAE scores,

which is done in order to be able to compare IAE scores for different step sizes.

In Figures 5.22 and 5.23, the controller responses in terms of deviation in molecular

weight, vM∆ , are given for disturbance rejection for 100% of positive and negative FR

changes used in experiments respectively.

Figure 5.22: Disturbance rejection responses of PID and MPC (+25 or +1.04 g/gmol step

disturbance on FR).

58

Figure 5.23: Disturbance rejection responses of PID and MPC (-25 or -1.42 g/gmol step

disturbance on FR).

It is seen from Figures 5.22 and 5.23 that IAE scores are 0.07 and 0.10 for positive input in

FR and 0.18 and 0.26 for negative input in FR, for PID and MPC respectively. Besides its

lower IAE scores, the settling times of the PID controllers are also smaller compared to MPC.

Disturbance rejection performances of the controllers for positive and negative disturbances

in temperature are given In Figures 5.24 and 5.25. The disturbance magnitudes studied are

the 50% of the T changes used in the experiments.

Figures 5.24 and 5.24 show that the controllers’ responses are similar to the cases where

disturbances are introduced to FR (Figures 5.22 and 5.23). The IAE scores of the controllers

are 0.08 and 0.16 for positive disturbance in T, and 0.21 and 0.22 for negative disturbance

in T, for PID and MPC respectively. The reason of larger settling times and IAE scores seem

to be the slower response of the extruder to the changes in temperature.

59

Figure 5.24: Disturbance rejection responses of PID and MPC (+10 oC step disturbance on T).


60

IAE scores and settling times calculated from Figures 5.24 and 5.25 are given in Table 5.10.

From Figures 5.22 to 5.25 and Table 5.10 it can be seen that both controllers are successful

in eliminating the effects of the disturbances, PID being a little better.


performances.

MPC PID

Case IAE Score Settling Time IAE Score Settling Time

Positive Disturbance on FR 0.10 1200 0.07 1000

Negative Disturbance on FR 0.26 700 0.18 550

Positive Disturbance on T 0.16 1200 0.08 1000

Negative Disturbance on T 0.22 900 0.21 500

The performances of the controllers for different step changes in the disturbances (50% of

the experimentally studied range in FR and 100% in T) are also investigated and the

responses are given in Figures 5.26 to 5.29.

When Figures 5.22 to 5.25 are compared to Figures 5.26 to 5.29, it can be seen that the

responses of the controllers, PID and MPC, changes linearly with the magnitude of the

disturbance. This conclusion can also be observed from the IAE scores given in Tables 5.10

and 5.11. These results are also in agreement with the assumption that the plant would

behave linearly in the studied ranges of the inputs. However, the settling times of the

controllers are not affected much by the magnitude of the disturbance (Tables 5.10 and

5.11).

61

Figure 5.26: Disturbance rejection responses of PID and MPC (+12.5 or +0.57 g/gmol

step disturbance on FR).

Figure 5.27: Disturbance rejection responses of PID and MPC (-12.5 or -0.71 g/gmol step

disturbance on FR).

62

Figure 5.28: Disturbance rejection responses of PID and MPC (+20 oC step disturbance on T).


63

5.5.5 Robustness Analysis

The robustness analysis is carried out to observe the performance of the controllers under

modeling errors. In order to change the plant behavior, the gains of the plant models are

changed by +10%.

Figures 5.30 and 5.31 show the controllers’ set point tracking performances for the 10%

increase and 10% decrease in model gains respectively. Normalized IAE scores calculated

from these figures are given in Tables 5.12 and 5.13. The IAE scores are normalized by

dividing with the magnitude of the change and corresponding time range.

It is observed from Figure 5.30 that MPC cannot achieve the set point due to the constraints

on manipulated variable for a relatively large negative set point value (-9050 g/mol).

However, PID achieves the set point by applying more control actions. Figure 5.31 shows

the case where the plant model gains are reduced by 10%. In this case, for the same set

point value, both controllers fail to achieve the set point although PID gives less offset by

keeping the screw speed on its lower limit.

In set point tracking, the controllers show similar responses in the case of modeling errors

(Figures 5.30 and 5.31) compared to the case without modeling error (Figure 5.21),

producing almost the same IAE scores (Tables 5.9, 5.11 and 5.12).

64

Figure 5.30: Set point tracking responses of PID and MPC (10% increase in model gains).

Figure 5.31: Set point tracking responses of PID and MPC (10% decrease in model gains).

65


increase in model gains).

Time range (s) 0 to

1000

1000 to

2000

2000 to

3000

3000 to

4000

4000 to

5000

5000 to

6000

PID 0.99 1.16 0.00 0.99 0.21 0.00

MPC 0.99 1.17 0.01 0.99 0.21 0.00


decrease in model gains).

Time range (s) 0 to

1000

1000 to

2000

2000 to

3000

3000 to

4000

4000 to

5000

5000 to

6000

PID 0.99 1.17 0.00 1.00 0.21 0.00

MPC 0.99 1.17 0.01 0.99 0.21 0.00

Figures 5.32 and 5.33 show the controllers’ responses to the disturbances in feed rate for

the case where the model gains are increased by 10%. Similarly, Figures 5.34 and 5.35

show the responses to the disturbances in feed rate where the model gains are decreased

by 10%. The IAE scores calculated from these figures are given in Tables 5.13 and 5.14.

66


disturbance on FR) (10% increase in model gains).

Figure 5.33: Disturbance rejection responses of PID and MPC (-1.42 g/gmol step

disturbance on FR) (10% increase in model gains).

67


disturbance on FR) (10% decrease in model gains).

Figure 5.35: Disturbance rejection responses of PID and MPC (-1.42 g/gmol step

disturbance on FR) (10% decrease in model gains).

68

Figures 5.36 and 5.37 show the controllers’ responses to the disturbances in temperature for

the case where the model gains are increased by 10%. Similarly, Figures 5.38 and 5.39

show the responses to the disturbances in temperature where the model gains are

decreased by 10%. The IAE scores calculated from these figures are given in Tables 5.13

and 5.14.

It can be seen from Figures 5.32 to 5.35 that the modeling errors do not have strong effect

on the disturbance rejection behaviors of the controllers. Although there are small

differences in IAE scores, PID and MPC respond to the disturbances similarly compared to

the case which there is no modeling error (Figures 5.22 to 5.25).


(10% increase in model gains).

69


(10% increase in model gains).


(10% decrease in model gains).

70


(10% decrease in model gains).

From Figures 5.30 to 5.39 and from Tables 5.13 and 5.14 it is seen that, both of the

controllers are robust and are capable of eliminating the effects of disturbances and tracking

the given set points even under the modeling errors.


performances (10% increase in model gains).

Case MPC PID

IAE

Score

Settling

Time

IAE

Score

Settling

Time





71


performances (10% decrease in models gains).

Case MPC PID

IAE

Score

Settling

Time

IAE

Score

Settling

Time





72

CHAPTER VI

CONCLUSIONS

In this study, experimental and theoretical studies are done to design an inferential control

scheme for the feedback control of RPET degradation during extrusion process. The basic

conclusions arrived are as follows:

• Degradation of PET is found to be highly nonlinear and it is affected by different

variables such as barrel temperatures and residence time spent in the extruder.

• The product quality is considered to be determined by molecular weight, Mv, and can be

measured indirectly by intrinsic viscosity ([η]).

• Dynamic experiments revealed that the extrusion process is highly nonlinear. It shows

different behaviors for the positive or negative deviations of the inputs.

• Convolution models are selected to be the best and to be used in simulations rather

than single or dual transfer functions of different properties.

• In the control system screw speed is evaluated to be manipulated variable using the

SVD analysis while barrel temperature and feed flow rate left to be probable

disturbances.

• In the control system designed, two different controllers ( +cG and −

cG ) are used for

positive and negative parts of the process model.

• PID and MPC controllers’ performances are found to be successful for set point tracking

and disturbance rejection cases.

• PID and MPC controllers are proven to be robust under modeling errors (+10%).

• The designed inferential control scheme can be utilized in controlling the product quality

in terms of molecular weight both with PID and MPC, once the off-line necessary data

for the estimator is obtained.

73

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77

APPENDIX A

MOLECULAR WEIGHT DETERMINATION

Viscosities1 of polymer solutions are usually measured in glass viscometers (Figure D.1), in

which the solution flows through a capillary tube by gravity. Two common types are Ostwald

and Ubbelohde viscometers. The Ubbelohde viscometer has the advantage that the flowing

solution is not affected by the amount of the liquid in the reservoir. Thus, the concentration

of the solution can be changed directly in the viscometer.

A.1 Molecular Weight Determination

The molecular weights (Mv) of the collected samples are determined by dilute solution

viscometry, using a Ubbelohde type viscometer.

0.06 gram of sample is weighted and dissolved in 6 ml of trifluoroacetic acid (TFA) in a test

tube, giving 1 g/dl concentration.

Initially, the flow time of pure solvent (t0) between two mark points is measured using a

chronometer, and recorded. The initial amount of pure solvent (TFA) in the viscometer is 7

ml. Measurement of t0 is repeated until same ‘minute:second’ value is read with the previous

flow (or flow time became constant).

1 “Since polymer solutions are non-Newtonian, intrinsic viscosity must be defined, strictly speaking, in terms of the

zero-shear or lower Newtonian viscosity. This is rarely a problem because the low shear rates in the usual

glassware viscometers give just that. Occasionally, however, extrapolation to zero-shear condition is required”

[Rosen, 1982].

78

Figure A.1: Ostwald and Ubbelohde viscometers.

When the flow time of pure solvent (t0) got constant in means of seconds, 1 ml of RPET-TFA

solution is added from the test tube to the viscometer, and the flow time of this new

concentration (t1) is measured, again until it got constant in means of seconds.

Above step is repeated 4 times (t1, t2, t3, t4) (see Appendix E). Using these flow times,

values for reduced viscosity (ηred) were calculated and plotted against corresponding

concentration ([C]) values. Then the best line (trend line)1 passing through these points is

calculated and its intercept at zero concentration is taken as intrinsic viscosity ([η]). After

the intrinsic viscosity value was obtained, Equation A.1 is used to calculate the Mv of the

sample.

[ ] αη1

=

KM v (A.1)

Symbols K andα refer to the Mark Hauwing constants for PET-TFA solution.

1 For any sample, when the ‘reduced viscosity’ was plotted against ‘concentration’, if the R2 value for the best line

(trend line) was below 0.98 that molecular weight determination was repeated for that sample. By this way, the

experimental errors were tried to be kept at minimum.

79

APPENDIX B

CALIBRATION FOR RESIDENCE TIME AND FEED RATE

B.1 Residence Time

Average residence time (ART) measurements for the studied screw speeds are given in

Table B.1. Same ART values were measured for different feed rates, as pointed in the

literature [Xanthos, 1992].

Table B.1: Average residence times for studied screw speeds.

Screw speed (rpm) Residence time (second)

50 255

125 137

200 122

275 115

350 15

425 13

500 11

In Figure B.1, it is seen clearly that ART changes in steps for screw speed. In other words,

for a range of screw speed values, ART remains almost unchanged.

80

Figure B.1: Avarage residence times measured at different screw speeds.

B.2 Feed Rate

The extruder control panel had the feed rate (FR) setting between the values of 0 and 999.

Thus, the feed rate and corresponding flow rate (g/min) values needed to be calibrated. The

calibration data for studied feed rate settings is given in Table B.2.

Table B.2: Flow rate calibration data for studied feed rate settings.

Feed rate reading Flow rate (g/min)

25 3.85

50 5.70

75 7.12

100 8.16

Figure B.2 shows that the relation between the FR and corresponding flow rate can be

accepted as linear.

81

Figure B.2: Feed rate settings and corresponding flow rates (g/min).

82

APPENDIX C

EXPERIMENTAL DATA

Table C.1: Molecular weight (g/gmol), Mv, change for different SS and FR values at 270 oC.

Screw Speed (rpm)

FR 50 125 200 275 350 425 500

25 (3.85 g/min) 7769 7873 10549 9268 10352 6313 9788

50 (5.70 g/min) 8361 669 12786 12769 6679 9828 12652

75 (7.12 g/min) 1696 14053 11741 5924 6183 10265 12665

100 (8.16 g/min) 12521 12226 9467 9146 13177 10221 11966


Screw Speed (rpm)

FR 50 125 200 275 350 425 500

25 (3.85 g/min) 6689 8774 12239 8520 9194 9913 7322

50 (5.70 g/min) 8642 10234 8416 10804 13042 13344 9587

75 (7.12 g/min) 10259 13103 10965 5932 8120 9748 10436

100 (8.16 g/min) 6809 9506 8572 6949 8355 10744 11310


Screw Speed (rpm)

FR 50 125 200 275 350 425 500

25 (3.85 g/min) 4504 3487 5507 2024 4805 5101 5584

50 (5.70 g/min) 7125 6121 7082 8642 7995 8956 5790

75 (7.12 g/min) 6034 11201 6890 9116 7218 6049 8680

100 (8.16 g/min) 9244 9250 7708 9331 6876 9346 7358

83

Table C.4: Mv (g/gmol) change with SS, FR and T (dynamic runs).

Time

(s)

SS

Plus

Step

(25 rpm)

SS

Minus

Step

(50 rpm)

FR

Plus

Step

(1.04 g/gmol)

FR

Minus

Step

(1.42 g/gmol)

T

Plus

Step

(20 oC)

T

Minus

Step

(20 oC)

0 11546 11301 11300 11300 11300 11500

10 11375 11371

20 10810 11051 11310 11555

30 11546 10505

40 10965 10445 11611 11042

50 11718 10271

60 11156 7969 11750 10520

70 11617 11868 9877

80 12200 6279 12019 9947

90 12495 12137 8610

100 12961 5405 12658 8977

110 13457 12382 7885

120 13928 3844 7618 13079 8511

130 14694 12548 5289

140 14388 3601 2576 13008 8015

150 14140 12538 1275

160 14106 2449 973 12675 7657

170 12495 1500

180 2147 2207 12632 6936

190 2259

200 1466 12628 5972

210

220 1764 7136

230

240 1796 8203

250

260 1696 8280

270

280 8292

84

APPENDIX D

PID CONTROL LOOP AND PROGRAM CODES

PID control loop designed using SIMULINK is given in Figure D.1.

Figure D.1: Feedback control loop for PID controllers designed in SIMULINK.

Code for the ‘SS+ Model’ is given below as an example for all positive and negative models.

% Screw Speed Dynamic Model (Positive Part)

function [ c_t_SS_Positive ] = SS_Positive_func(input)

delta_input = input(1);

delta_time = input(2);

try a_SS_Positive=evalin('base','a_SS_Positive'); catch a_SS_Positive = [];

85

end;

try delta_m_SS_Positive=evalin('base','delta_m_SS_Positive'); catch

delta_m_SS_Positive = []; end;

model=evalin('base','SS_Positive_Fit');

n = delta_time+1;

a_SS_Positive(n) = model(delta_time);

delta_m_SS_Positive(n) = delta_input;

if n >=2

count = numel(a_SS_Positive);

a_vector = a_SS_Positive(2:count);

m_vector = delta_m_SS_Positive((count-1):-1:1);

summation = a_vector.*m_vector;

c_t_SS_Positive = sum(summation);

else

c_t_SS_Positive = 0;

end

assignin('base','a_SS_Positive',a_SS_Positive);

assignin('base','delta_m_SS_Positive',delta_m_SS_Positive);

Code of the model predictive control part is given below.

% MPC code

clc;clear

addpath Source;

load Fitted_Models;

load MPC_Positive_Minus;

Stop_Time = 500; %Total Simulation Time

delta_input = 0; %maginitude of delta input

delta_input_time = 0; %time of delta input

Set_Points = [0 0 -10 0]; % Set Point(s) vector with 'n' elements

Set_Point_times = [0 1000 2000 3000]; % Set Point Time(s) vector with 'n'

elements

%Disturbance Types -> (0) None || (1) FR Positive || (-1) FR Minus || (2) T Positive

|| (-2) T Minus

Dist_Type = 0;

Dist_Magnitude = 0;

Dist_Time = 0;

% Initial Values

MPC_Positive_state = mpcstate(MPC_Positive);

MPC_Minus_state = mpcstate(MPC_Minus);

u = 0;

u_prev = 0;

du = 0;

d_Positive = 0;

d_Minus = 0;

Mv_t = 0;

Dist_prev = 0;

% Simulation starts here

for time = 0:1:Stop_Time

86

for sp_detect = 1:numel(Set_Point_times) % Follow Set Points

if time == Set_Point_times(sp_detect)

Set_Point = Set_Points(sp_detect);

end

end

if not(Dist_Type == 0); % Disturbance Part

if time >= Dist_Time

dist_delta_time = time - Dist_Time;

Dist_delta_u = Dist_Magnitude - Dist_prev;

Mv_dist = disturbance_func(dist_delta_time,Dist_Type,Dist_delta_u);

Dist_prev = Dist_Magnitude;

else

Mv_dist = 0;

end

else

Mv_dist = 0;

end

region = region_select(d_Positive,d_Minus,Mv_t,Set_Point,Mv_dist);

switch region

case 0, % No models need to run

case 1, % SP Positive Region

clear minus_start_time

try Positive_start_time;

catch Positive_start_time = time;

end

u = mpcmove(MPC_Positive, MPC_Positive_state,Mv_t,Set_Point,[]);

du = u - u_prev;

Mv_t_prev = Mv_t;

delta_time = time - Positive_start_time;

Mv_t_Positive = SS_Positive_func([du delta_time]);

Mv_t = Mv_t_Positive + Mv_dist;

u_prev = u;

d_Positive = Mv_t - Mv_t_prev;

if abs(d_Positive) < 0.000001

d_Positive = 0;

end

if abs(Mv_t) < 0.1

Mv_t = 0;

end

assignin('base','Positive_start_time',Positive_start_time);

case -1, % SP Minus Region

clear Positive_start_time

try minus_start_time;

catch minus_start_time = time;

end

u = mpcmove(MPC_Minus,MPC_Minus_state,Mv_t,Set_Point,[]);

du = u - u_prev;

Mv_t_prev = Mv_t;

delta_time = time - minus_start_time;

Mv_t_minus = SS_Minus_func([du delta_time]);

87

Mv_t = Mv_t_minus + Mv_dist;

u_prev = u;

d_Minus = Mv_t - Mv_t_prev;

if abs(d_Minus) < 0.000001

d_Minus = 0;

end

if abs(Mv_t) < 0.1

Mv_t = 0;

end

assignin('base','minus_start_time',minus_start_time);

case 10, % Disturbance Positive Region

case -10, % Disturbance Minus Region

end

Mv_hist(time+1) = Mv_t;

Mv_dist_hist(time+1) = Mv_dist;

u_hist(time+1) = u;

du_hist(time+1) = du;

SP_hist(time+1) = Set_Point;

region_hist(time+1) = [region];

MPC_Dist_Response(time+1,:) = [time Set_Point Mv_dist Mv_t u];

end

disp 'finished'

subplot(2,1,1);plot(Mv_hist,'k');hold on;plot(SP_hist,':r');plot(Mv_dist_hist,'r');

subplot(2,1,2);plot(u_hist)

figure

plot(region_hist)

88

APPENDIX E

SAMPLE CALCULATIONS

E.1 Molecular Weight Calculations

In Table E.1 an example of the experimental data collected during the viscosity

measurements are given.

Table E.1: Experimental data (an example).

Sample: steady-state experiments (T: 290 oC, FR: 25, SS: 50 rpm)

0.0630 grams sample + 6 ml TFA

Solution in

reservoir tn

Measured flow times

(min.sec.milisec) Constant flow time (sec)

Pure Solvent{7 ml} t0 1) 2.28.12

2) 2.28.26 148

+ 1 ml addition t1 1) 2.37.91

2) 2.37.57

157

+ 1 ml addition t2

1) 2.47.31

2) 2.46.72

3) 2.47.09

166

+ 1 ml addition t3 1) 2.55.38

2) 2.55.88 175

+ 1 ml addition t4

1) 3.03.16

2) 3.02.56

3) 3.02.17

182

+ 1 ml addition t5 1) 3.09.72

2) 3.09.19 189

89

Using the measured flow times, calculations are done and results are given in Table E.2.

Table E.2: Reduced viscosity for different concentrations.

n

Flow time of

solution ( 0t )

Relative viscosity

0

0

t

ttn

rel

−=η Concentration

in reservoir [C]

Reduced Viscosity

[ ]C

rel

red

ηη =

0 148 0 0.0000 -

1 157 0.061 0.131 0.463

2 166 0.122 0.233 0.521

3 175 0.182 0.315 0.579

4 182 0.230 0.382 0.602

5 189 0.277 0.438 0.633

The calculated values for reduced viscosity (ηred) were plotted against corresponding

concentration ([C]) values, as illustrated in Figure E.1. Then the best line (trend line) passing

through these points was calculated and its intercept at zero concentration was taken as

intrinsic viscosity ([η]).

The viscosity average molecular weight (Mv) is calculated using Equation E.1.

[ ] αη1

=

KM v

mol

g

xM v 8.6689

104.1

393.05625.1

3=

=

−

(E.1)

90

Figure E.1: Reduced viscosity change with concentration.

E.2 PID TUNING CALCULATIONS

The PID controllers studied are first tuned by modified Ziegler-Nichols method as given in

the literature [Seborg et al., 1989]. They are then further fine tuned by trial-error procedure.

Table E.3 gives the ultimate gains (Ku) and the ultimate periods (Pu) of the continuous

oscillations for the positive and negative PID-model pairs.

Table E.3: Ultimate gains and ultimate periods of positive and negative PID-model pairs.

Pair Ku (rpm.gmol/g) Pu (seconds)

PID+-SS+ 0.00748 182

PID--SS- 0.0174 112

The relations between the Ku and Pu and the PID parameters ( DicK ττ ,, ) are given in

Table E.3.

91

Table E.4: Original and modified Ziegler-Nichols settings for PID controllers [Seborg et al.,

1989].

cK iτ Dτ

Original (1/4 decay ratio) 0.6Ku Pu/2 Pu/8

Some Overshoot 0.33Ku Pu/2 Pu/3

No Overshoot 0.2Ku Pu/2 Pu/3

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